I thought that "self defense" was something you had to prove yourself, where you've admitted to committing a crime (killing someone) but that you had a specific circumstance that mitigated the charge.

I assumed it was similar to a "not guilty by reason of insanity" plea, where you admit you committed a crime but there are special circumstances (that you have to prove) that mitigate it. You can't just stay silent the entire time and force the prosecutor to prove you were insane - you have to actually provide some evidence you were insane.

Is this not true?

That's correct, it's an affirmative defense. You admit to the crime, but assert that you were privileged to take an otherwise illegal action.

As I understand it, it differs depending on state law.

This is totally possible—if Rosenbaum had killed Rittenhouse, he would have been laughed out of court on a self-defense claim.

People claim self-defense all the time, it rarely works. In this case, where the clear, legally defined standard of self-defense was met, Kyle walked.

Did you follow the (nearly simultaneous) trial of the killers of Ahmaud Arbery? That prosecutor didn’t seem to have too much trouble at all proving that particular negative — those guys are all going away for life.

That case would be the equivalent if Grosskreutz killed Rittenhouse and tried to claim self defense.

An alternative equivalent would be if Arbery was successful at taking away the firearm and killing his attackers.

What? Not at all. Arbery attempted to gain control of the gun and was killed, and the defense attempted to claim self-defense. I was responding to the claim that self-defense is some kind of automatic not-guilty verdict. It evidently is not.

Arbery, like Rittenhouse, was not the aggressor. Rittenhouse had a firearm to defend himself, Arbery did not. If Arbery was successful in taking the firearm from his assailants, self-defense would be a very viable claim.