The problem with this mess is that the police on their own decided that Zimmerman acted in self-defense and closed the case. What should have happened, and I think what would have happened in most of the U.S., is that the police would say, "You shot this guy? You're under arrest." The police don't need anything more than probable cause to arrest somebody, and a dead body, a gun and a confession would be plenty. Then the D.A. would investigate and decide whether to press charges, and if so what charges under what laws. If it went to trial the defendant would have the opportunity to tell the jury that he was in fear for his life and that his fear was reasonable under the law, and the jury would decide whether he was right. The cops shut down the process before it even began.
Small detail, which has been gone over before in other threads, *just* making an initial contact statement to the police, in which you describe your actions, and your reasons, is not a 'confession'. Some of what you say could be used later, either as evidence, or to undermine credibility, but 'I shot him, it was self defense' really doesn't rank as a confession.
And in the case of the difference between being charged with murder, and being released as self-defense, the narrative of actions will often be the same, i.e. 'I shot him'.
Sometimes people can't stop blurting out their guilt at the scene as in 'You bet I shot the punk, he had it coming, he looked at me funny', but generally confesssions are sworn statements taken after being Mirandized, admitting to committing a crime, which are recorded and signed with witnessess affidavits.
Zimmerman didn't give a confession.
ETA to repeat the obvious: This in no wise advocates that any particular person *should* give any statement to the police, or that this implies anything about the shooter, or the victim.