Morally he didn't do it, and maybe Quirrel even had a desire to kill her sitting on a back burner before Harry got involved, but her death was caused by her interaction with Harry. It is no stretch to say that there is at least one hypothetical sequence of actions Harry could have taken, even given knowledge at the time (not realizing she worked for Lucius or was an animagus) which would not have resulted in her death. Heck, doing nothing would have resulted in her not death.

That is the level of challenge Harry is taking upon himself. Not just to not kill anyone, not just to keep your hands clean, not just to save people when he can. He's declaring that if any innocent person anywhere dies and there's something he could have done differently to save them, that's his failure condition. You can't do that.

That said, I thought about it a few minutes more and it could be his resolution is really about knowing he doesn't know how bad the situation is. It's certainly possible to get through, say, a political power struggle with someone like Lord Malfoy without anyone getting killed. Harry considers it possible but doesn't yet believe that his opponent is Voldemort. If his opponent is Voldemort avoiding casualties is impossible. If his opponent is someone less evil (though still pretty nasty), and the scope of the conflict is much smaller, he might be able to pull it off.