1. Harvey Dent's state of mind
At the time of his conversation with the Joker, Dent is seething with rage about Rachel's death. He's not really thinking pragmatically.
Don't forget that the Joker swapped Harvey and Rachel's locations when he spilled the beans. He expected that Batman would rescue one, and send the police to rescue the other.
Joker also correctly predicted that Batman would try to rescue Rachel (he saw Batman dive out of a window to save her before, it makes sense that he'll try to save her again).
Harvey resents Batman for saving him and not Rachel. It's somewhat expected that Batman has a better shot at saving someone compared to the police, and since Harvey wants Rachel to be saved instead of him, Harvey resents Batman for not trying to save Rachel instead of him (Harvey does not know that the locations were swapped).
Notice that while Harvey still is in the hospital, no one has told him about the locations having been swapped. It would only make Harvey feel even worse (knowing that Rachel could have been the one that was saved, but died anyway). The Joker manages to have his conversation with Harvey before he is aware of the swapped positions.
Also keep in mind that Rachel and Harvey were part of this game because corrupt police officers captured them. Harvey is well aware of this, since he remembers being taken, and likely has already heard that the same happened to Rachel.
2. The Joker's arguments
Joker's argument to Harvey relies on a few key points, some of which are implicit (relying on what Harvey already thinks, without the Joker explicitly having to state it). I'm describing these from the point of view of the Joker, based on what Harvey thinks is the truth:
- I am just one man. I could not do this by myself. It is Gotham that enables these atrocities to happen.
- This atrocity happened because the people you trust most are corrupt.
- The corrupt police captured Rachel. Batman came to rescue you, not Rachel. The police failed to rescue Rachel. Had they not been corrupt, Rachel wouldn't have needed to be saved in the first place.
- Your enemies are corruption and organized crime. I am not corrupt (as I am not on the side of the good guys), and I am most definitely not organized (because I'm a loose cannon).
- We are fighting the same enemy. And that enemy is the belief that we can/should control everything and keep everyone safe. We can't. Even with all the resources they had available, the police and Batman could not save Rachel.
- Everything is chaos. Chaos is the most natural state of being. We are all subjected to it. I am merely showing you that it exists; even if I'm not here. I don't create chaos, I just point it out.
The emphasis is on the people who Harvey would likely consider equally responsible for Rachel's death. Notice how Joker's arguments mostly put the blame on the corrupt and fallibly good guys. Harvey is looking for a guilty party to convict, and the Joker is subtly offering the police and Batman as the biggest perpetrators of the injustice of Rachel's death.
These are subtle points, mostly driven by what Harvey already knows, and slightly tweaked by the Joker's argument that he's not the only evil.
Harvey realizes that the Joker is not the sole reason why Rachel died. If anything, the fact that Harvey survived proves to Harvey that Rachel could have survived this too.
But she didn't. Because they (police+Batman) failed to save her. They contributed to Rachel's death as much as the Joker did.
That's what Harvey takes away from his conversation with the Joker.
3. Harvey still considers the Joker to be guilty
During the Joker and Harvey's conversation, Two-Face is born. I won't call him Harvey anymore, because Two-Face operates under a different belief than Harvey Dent did.
The Joker is already convinced that he has swayed Two-Face, because he gets Two-Face to aim a gun at him.
However, the Joker was not aware of Two-Face's "coin flip" method. The coin flip is Two-Face's way of punishing the guilty. He does not outright kill everyone who was involved. He gives everyone a 50/50 shot.
This is the core of the answer:
Dent believes the Joker that others are also responsible. Dent does not believe that the Joker is in no way responsible. Dent forces the Joker's life to rely on "the coin flip", which proves that Dent considers the Joker to be guilty.
Two-Face considers everyone (who ends up getting a coin flip from him) to be part of the same conspiracy that killed Rachel. Whether they are part of it intentionally (Joker, the corrupt cops, Maroni) or unintentionally (Gordon, Batman) doesn't really matter to Two-Face.
Although we do not see Joker's 50/50 shot, we see Harvey flip the coin while still having a gun to the Joker's head, and we see the Joker still alive in a following scene. Logically, the Joker must have had a lucky coin flip.