To answer your last question first, and directly:
I don't know what the exact words of the original prophecy were. I don't think anyone does because they are not contained in any canon material (that I know of).
There are some non-canon sources that speak of the prophecy, but not only are they officially classified as "legend" material, it's not clear, even within those sources, whether the prophecy being recounted is truly the original prophecy or simply a retelling (and possibly a reinterpretation) of whatever the original prophecy was. For purposes of this discussion, it works just as well and is safe to assume that the original words have been lost to the mists of time, and only the general idea survived until Anakin's day.
The root of your confusion
That said, your larger question reveals many misconceptions of the prophecy and the greater story of Star Wars. I feel that you're asking the wrong question because you're starting with the wrong assumptions. By clearing up some of those misconceptions, I think you will find that most of your concerns and confusions disappear into irrelevance.
The problem is that your understanding of the Force itself and of the prophecy, just within the context of the films, is all wrong - but don't be offended, they are very common misunderstandings. In fact, that's why I'm taking the time to write out this long explanation: to put all those misconceptions to bed once and for all, for all the many people who are confused or outright mistaken in their understanding.
What is the Force?
The Force is not made of two sides. It is not made of Jedi and Sith. It is not made of dark and light. "Jedi and Sith", "Light and Dark" are ideas and concepts and they are ideas and concepts created by multitudes of Force Users over the millenia to describe how they feel about, or perceive the Force, or its users, or certain actions.
The Force simply is. It's a Force of nature (literally). "Good" and "evil" are human (or other species') inventions. They are labels put to actions of individuals, based on personal or cultural morality. In essence, they can be relative, they can be different from individual to individual, they can change with time and context, and they can be all-around ambiguous.
What makes up the Force? Does the Force have "sides"?
The Force is no more made up of "good" and "evil" than a stone is. It is only the consciousness that uses the stone to act that creates something good or something evil. I could use a stone to smash someone's skull, or I could use it to build a house, or a wall, or a bridge. The Force is no more made up of "good" or "evil" than water is. I could give you a glass of water, I could bathe you, I could water your crops, or I could drown you. But we'll talk a lot more about the Force as a flow of water a little later.
Even if we consider this supposed duality of Jedi and Sith, or light and dark, your initial assumption falls apart. It is, in short, a false dichotomy. According to the extended canon of the Star Wars Universe, there have been many practitioners and groups and belief systems of both the light and dark sides of the Force. The Jedi were simply one particularly successful cult of light-side users, and the Sith of the dark side. But the fact remains, that not all light-side Force users are Jedi, and not all dark-side users are Sith.
Just like you can be a badass MMA fighter, but not have a black-belt in Judo, a Jedi is simply a light-side Force user that has been granted an official title within the particular order and style of the Jedi. In the original trilogy, it wasn't even clear that the Emperor was a member of the Sith (at least going strictly by the movie canon only) until the Prequel trilogy came out. Similarly, in the sequel trilogy starting with Episode 7, there is no indication that Kylo Ren or Snoke consider themselves to be Sith. Neither, so far, use the title "Darth". Everything points to Vader's assassination of the Emperor, and Vader's subsequent death, as being the end of the rule of two, and the end of the Sith legacy. And yet, Kylo Ren and Snoke (and potentially the Knights of Ren, if they still exist), are still clearly dark-side Force users (by that narrow binary view of the Force). Chirrut Îmwe, in Rogue One, also seems to be a force-sensitive who is expressly described as not a Jedi.
Even if we consider the broader categories of light and dark as two sides of the Force, we'd still be wrong. There are apparently some Force users that would be considered "grey Jedi". Both Count Dooku, and Qui-gon Jinn his apprentice, could have been considered to be among them. Dooku, in fact, left the Jedi Order, but continued practicing as a Force user, neither of the light or of the dark, until later falling to the dark side.
Does bringing balance to the Force have anything to do with Sith and Jedi?
Yes and no. I'll come back to this, but let's focus on the "no" for now.
So the point of all this, is that your understanding of the prophecy of bringing "balance" to the Force is all wrong, because you don't understand what the Force is, what the Force is made of, or divided into, and thus you don't understand what it is that needs to be balanced. When you assume that "balancing" the Force has something to do with the number of Jedi and the number of Sith, then you are conflating "the Force" with "Force users".
If I can make a strained metaphor, imagine you have a party with 5 girls and 4 guys and the pizza order arrives. Someone says, "we need to divide up the pizza so everyone gets a fair amount". Is your solution to then kick 1 girl out of the party? And when there are just 4 girls and 4 guys left, then you say, "ok, now the pizza is fairly divided"?
You see how you missed the point of the Force? The Force is the pizza. The balance that needed to be found in that situation was with regards to the pizza, and had nothing to do with the gender of the party-goers. In fact, I could have chosen to "balance" the party-goers by way of their age, or by their height, or by their ethnicity, or by their clothing, and any way I would have gone about it would still have been wrong, because the focus needed to be on the pizza, not the party-goers.
The prophecy did not say that the Chosen One would "bring balance to the Force users". No, the Chosen One would bring balance to the Force itself. That means something was wrong, or out of balance, with the Force itself!
So what exactly does it mean to "bring balance" to the Force?
The Force doesn't have two sides, but it does have many facets. But even those facets are more reflective of the kind of person that the Force user is, rather than a reflection on the Force itself. There is no good Force and bad Force, there are simply Force users that choose to do "good" deeds by using the Force, and others that choose to do "bad" deeds by using the Force.
So the Force doesn't have two sides, it actually has many sides, but even those sides can't definitively said to be "good" or "bad". The Force simply is, the same way the universe simply is, and the same way that you can't say the universe is "good" or "bad". So now you're getting frustrated: what the hell are we balancing?
The answer comes from Lucas himself (but apparently the original recording is only available on the VHS of the Special Edition of A New Hope [Episode 4]). If the Force just is, it exists independent of any Jedi, any Sith, or any Force user. In such a state, it could be said to be in its "natural" state. The Jedi view the Force, and its power, and its will, as a river (there's that water comparison I promised you was coming). This is alluded to when Obi-wan says to "feel the Force flowing through you".
The Jedi and the metaphor of the river
A Jedi also views himself as a servant of that river, swimming in the river itself. Though a person swimming in a strong, deep river may be able to slow down slightly, or perhaps even pause for a bit on a rock or in an eddy, or sometimes swim from bank to bank, they can never swim back against the current of the river, and they can never choose to avoid their ultimate destination, or go anywhere that the river does not go.
Just so is the relationship between a Jedi and the Force: "You mean it controls your actions? Partially, but it also obeys your commands." When Obi-wan tells Luke to "let go" he is telling him to stop fighting the river, and let it take him where he needs to go.
Lucas also describes the relationship between the Jedi and the Force as a mutualistic symbiotic relationship (a relationship paralleled in the flesh by the symbiotic midichlorians). The power and the speed of the Force at a Jedi's back gives him or her his power. In fact, a Jedi who is not "listening to the will" of the Force, and fighting the current so to speak, is weaker.
At the same time, as an agent of the connections between all life, the Force guides Jedi to where they can connect with other life and build stronger bonds between them. The river of the Force contains all living things. "It surrounds us and penetrates us";"it binds the galaxy together." Both sides benefit in this relationship. The Force grows strong with more life, and the life that flows with the will of the Force also grows stronger. The Jedi live in harmony with the river, taking only what they need to survive, and at the same time protecting all the life of the river and the river itself.
The Sith and the river
But what about the Sith? Whereas the most important concern for a Jedi is the will of the Force, the most important concern for a Sith is themselves. Whereas a Jedi seeks to make the Force stronger, the Sith seeks to make himself or herself stronger. Whereas a Jedi tries to listen to the Force, the Sith wants the Force to listen to them. Whereas a Jedi swims in the flow of the river, the Sith seeks to control the flow of the river, to bend it into new directions where it never intended to go, and at times, to completely dam its flow and completely harness the power of its currents for their own purposes.
And just like a dam in the real world, though it brings great power to the owner of the dam, it completely devastates the ecosystem of the life that lives in and around the river. Fish are caught off from their spawning and feeding grounds, sediment builds up, lands upstream are flooded, fresh water ceases to flow downstream and lands dry up. The natural balance of that ecosystem, established over thousands of years, is completely upended.
A Sith is not a mutualistic symbiote with the Force, but rather, a cancerous parasite. Whereas Yoda describes the Force as his friend and "ally" - an equal and partner - for the Sith the Force is simply a tool, or even worse, a slave, for their own selfish designs and purposes. In his or her hate-filled use of the Force, a Sith destroys life and the connections between life, and thereby destroys the Force itself. The Sith user becomes stronger by gathering up all the Force power for himself, but in so doing makes the Force weaker and dimmer for everyone else. Whereas a Jedi would never draw more power from the Force than they need, would never draw more power from the Force than what it offers up, would never draw power from the Force at the expense of others, a Sith forces the Force to take power from others. And so the river downstream dries up, the current slows down, and the connections between life weaken.
This is exactly what Yoda was talking about in the prequels when he said "the dark side clouds everything" and when Mace Windu commented that the Jedi power to use the Force had "diminished". Palpatine (the Emperor, a.k.a. Darth Sidious) was such a strong Sith, and had gathered so much of the river's power to him, that the Jedi were left high and dry, unable to hear clearly the will of the Force anymore, unable to swim in its shallow waters, unable to accurately sense the future, nor the fact that the Sith were literally right under their noses. Yoda knew what was happening, and knew that the Sith had returned precisely because of the Jedis' weakened connection, but even he couldn't stop it: "Only the Dark Lord knows of our weakness".
The Jedi and the prophecy
This then, was the goal of the Jedi: to maintain the natural state of the river of the Force. They saw themselves as servants, caretakers, friends, and guardians of the Force and all the life and creatures that both gave the Force its power and in turn received power from that river.
Now we can finally understand why I said "yes and no" to the question of whether the Sith and Jedi have something to do with the prophecy of bringing balance to the Force. The answer is "no", because the prophecy is not directly about Sith or Jedi - it is instead about the health (or "balance", or in the French translation, "harmony") of the Force. The focus, as in my bad pizza metaphor above, is on the state of the Force itself. However, we can also say "yes" in that the Jedi and Sith are indirectly involved in the balance of the Force, just as the party-goers are involved in the eating the pizza, and just as we humans can affect the health of our ecosystem by our actions and behavior.
The Jedi followed a mantra similar to a responsible hiker or camper: "leave nothing but footprints." Even in their "use" of the Force, their goal was to leave the state of the Force as much unchanged as possible, almost as if they had never been there, and to oppose those who wished to change it. They would go along with the will, or flow, of the Force, even though you might not be able to characterize that will as good or evil, necessarily. It was simply a force of nature. At times, the will of the Force might even include the destruction of life, either as retribution, or as a sacrifice, especially when that destruction might allow other life to flourish.
In contrast to themselves, the Jedi saw the Sith as a destructive corruption of the river's natural state, and it is the Sith's selfish use of the Force, twisting it to their own desires, and collecting tremendous amount of power at the expense of everyone else, which result in an unbalanced and unnatural state.
Whereas the prophecy of balance is not specifically about the Sith (nor the Jedi), the inevitable conclusion that must be reached is that the mere presence of a dark-side user, particularly an ultra-powerful Sith, disrupts the normal balance of the Force and should be corrected; and the Jedi, as the guardians of that balance, see themselves as the agents responsible for making that correction.
So when a boy with an incredible affinity for the Force, possibly created by the Force itself, shows up, at the same time that the Sith reappear after a thousand years missing, and a dark, sinister, mysterious power begins to creep into and cloud the mind of all the Jedi, it is no wonder that many Jedi began to think of the prophecy of the one who would return the Force to its natural uncorrupted state.
In turn, when Revenge of the Sith (Episode 3) ends, and the Jedi are all but dead, the "balance" is definitely not restored. In fact, quite the contrary. The Sith Lord has attained his fully revealed, completely unchecked power ("unlimited power!"). He has built a dam so strong that he was able to defeat all the Jedi, to betray them without them knowing, and to turn their strongest against them. The river of the Force is now basically dry, and its former guardians and protectors all dead or on the run. Obi-wan makes this clear when he tells Anakin, "you were supposed to bring balance to the Force, not leave it in darkness."
A certain point of view?
Of course, all of this depends on the viewpoint of the Jedi, and as Yoda himself said, "a prophecy that misread could have been." So possibly, from the Sith point of view, if they even believed in such a parallel prophecy, the end of Episode 3 could very well have been some fulfillment of their understanding of "balance".
But from the point of view of the good guys, Anakin does actually fulfill his intended prophecy. When he kills the Emperor in Episode 6, he also turns back to the good side and therefore "kills" Darth Vader. With both Sith "dead", the dam is broken, the river of the Force is unleashed again and free to flow without hinderance, and the process of healing and a return to a natural state can begin.
(Episode VII's title of "The Force Awakens" seems to speak exactly to that idea [which is cool], but some might ask why it would take thirty years for the Force to wake up? Ignoring the fact that the plot, and title, and time difference between Episodes VI and VII were not part of Lucas' original story arc, it would be easy and sensible to rationalize that, just as a scarred and polluted and parched landscape might take several decades or even centuries to fully recover from an environmental disaster, so might the restoration of the balance of the Force be a process that requires a significant amount of time.)
The "Return of the Jedi" is not just Luke becoming a true Jedi by letting go his selfish fears, following the will of the Force, and confronting Vader, but it is much more importantly Anakin's return to the way of the Jedi and his fulfillment of the original prophecy. Episode 1 through 6 are really the story, the journey, and the prophecy of Anakin.
(Luke can definitely be considered an important catalyst to the fulfillment of the prophecy, but it is Anakin's actions and specific choice, that directly end the Emperor's, and his own, tyrrany over the galaxy and the Force. Again, as Episodes VII - IX were not originally part of the story arc, it may be necessary to "misinterpret" the prophecy again. If the process of restoring balance to the Force requires more time and effort, then perhaps Luke has a larger role to play in the prophecy. This complicates matters, however, as the prophecy speaks of a Chosen "One" [not "Two"]. Either we then consider Luke's role to be separate from Anakin's [i.e. not involved in the prophecy directly as the prophecy is already fulfilled] or we can speculate that as Luke is Anakin's son, the product of his loins, the inheritor of his powers, and the result of his crucible by fire, that any of Luke's actions in regards to the restoration of the Force are still attributable to Anakin [and then we consider that Anakin only began the process of fulfilling the prophecy, which his son, or other progeny, will then complete.] We'll have to wait for this new trilogy to finish before we make any definitive guesses [ha] as regards to how the prophecy may or may not be related to it.)
Thematically, narratively, and symbolically, the end of Episode 6 must be the realization of the "balance" that the Jedi hoped to restore. And if you don't want to take the word of the Jedi, nor of the Sith, consider it from the viewpoint of the Force itself (if the Force can even have a view). Would the Force rather be, just be, a Force of nature that does its thing and has some dudes (Jedi and others) that help make sure the Force can continue to just be, or would the Force rather be a slave to the plans of one dude (a Sith) who wants to control everything?
From my point of view, I think the answer is pretty obvious.
(Episode 8 update: I thought I would have a lot more to update here after The Last Jedi, but that was such a disappointing movie, perhaps the worst Star Wars movie ever, and did very little to add anything worthwhile to the overall mythos of the story, of the Jedi, and of the Force. The only thing I can say is that it does reinforce the idea that the titles of Jedi and Sith are immaterial to the will or balance of the Force. Even if no Sith exist, there are those who will inevitably succumb to their own selfish desires for power and seek dominion over the Force itself and over other life. Even if no Jedi exist, there are those who inevitably hear the calling of the Force and rise up to see its will carried out and to take care of the life that it represents.)