"He wanted to kill Dumbledore" would have been poor dramatic phrasing. "The response to compulsion was killing" could mean that he has a memory and history of this, or it could simply mean that in his state of mind, killing seems like the natural response to being compelled to do things by others. If I were trying to write that, I would sooner write "the response to compulsion was killing" than "He wanted to kill Dumbledore."

The fact that Harry underwent a serious personality change on exposure to the dementor, and Hermione speculated that such a thing might happen to a person who already had that darker personality within them, is a substantial piece of evidence that Harry has something more unusual going on than some personality quirks. The phrasing used in that scene, on the other hand, I do not think can reasonably be treated as evidence of anything in particular. In fact, I can't think of a single explanation for Harry's personality change which would make the phrasing seem weird, given that artistic impact of the words being used is as important a consideration as their connotations.