Who is the "One" in The Matrix?

Question

One of the conversations between Morpheus and Neo in The Matrix is:

When the Matrix was first built, there was a man born inside who had the ability to change whatever he wanted, to remake the Matrix as he saw fit. It was he who freed the first of us, taught us the truth - As long as the Matrix exists, the human race will never be free. After he died, the Oracle prophesied his return and his coming would hail the destruction of the Matrix, end the war, bring freedom to our people. That is why there are those who have spent our entire lives searching the Matrix looking for him.

Morpheus believed that Neo is the one, but Neo wasn’t born inside the matrix. He was harvested in the machine world and jacked into the matrix right from birth.

So is he really the "One"?

Google search results of Who is the "One" in The Matrix? return "Agent Smith" as answer, but how can he be the One?

Answer 1

He's never named, but since he is an intentional feature of the Matrix introduced by the machines it almost doesn't matter.

The quote is from Morpheus, who along with the rest of the human race has been heavily mislead by all of the machines, especially the Oracle. We have no way of knowing if the person mentioned even existed let alone what his name was.

Answer 2

What the Architect explains is, the Matrix is in the 6th iteration, and with each iteration (from the 3rd onward), the code that allows human choice produces a "The One". The One tries to fight the system, fails, his code gets assimilated into the Matrix willingly (as the alternative is the threat of human extinction), the system is rebooted into a new version, and after Zion is razed, The One picks 23 humans from the Matrix, and leads them to Zion.

Trying to fit Morpheus knowledge with the Architect explanation would mean Morpheus is referring to the The One from the 5th Matrix. For Zionists, there was only ever 1 Matrix as they have no way of knowing what came before. For Zionists, The One was born inside the Matrix, he was quite literally part of the Matrix since the reboot - because The One doesn't tell them about the previous versions of the Matrix, and by the time a new The One surfaces he has already been long dead.

Neo breaks this loop, because he fell in love, and his emotions override his logical thinking. He rather risk every human to save Trinity, than do the logical thing that'd be saving everyone else. The architect says he is overlooking the fact, if humanity dies, Trinity is dead anyway, but he knows Neo will pick Trinity anyway.

Another way of looking at this is Mr. Smith is The One (the final The One), because he fits the description equally well. He has part of Neo's code imprintend in him, he was born out of the machine, he almost completely destroys the Matrix, he forces the hand of the machines into accepting truce and ending the war (as the only way to stop him), and by consequence humans and machines are free to choose if they want to leave the Matrix or not.

Answer 3

The One, as Morpheus mentions, lived way back in the past, was born inside the system, and had the power to change everything else and all that Morpheus described. But he died too.

Neo or Agent Smith is not the one.
The One is unnamed in The Matrix Trilogy.

Now who is Agent Smith?
He is just an agent who is ordered to look out for instability/anomaly in the system. For controlling that, he also has some super power. Then he goes rogue, like a rogue program, and need to be terminated. Neo kills him but his super power enables him to return back.

Who is Neo?
Neo is the sixth incarnation of The one, who died long ago. But as you said,

Morpheus believed that Neo is the one.

But how?
Neo is linked to Agent Smith. So, if anyone can achieve that superpower which allows Smith to run around, it is only Neo.

However, in the sequels, Smith is revealed to have been linked to Neo, which enabled him to resist being sent to the system source where destroyed programs are permanently deleted.

So, is Neo the One?
In a way, yes. He is the only one with access to Smith's power, so he must be brought into the system.

This link and this link provide beautiful explanations and analogies, especially the first one.