Lennon's assassin spent weeks listening to Lennon records, weeping and tearing his hair at the fakeness of his idol. Loughner had a thank you letter in his safe (in which he had documents that spoke of “assassination”). The note was from Giffords concerning an event he'd attended.
What sticks in my mind (thanks to Graham Harman for drawing my attention to this) is that at the event, Loughner had asked Giffords some kind of question like “How do you know words mean anything?” Giffords had kind of brushed him off (I believe she spoke to him in Spanish and moved on).
Remember that scene in Twelve Monkeys where the guy with the virus in the test tubes asks a provocative question slightly out of left field to the lecturer—he needs to have his fantasy confirmed.
I'm still betting my bottom dollar that rather than thinking Giffords was possessed by demons (a schizophrenic paranoia), Loughner thought that Giffords was a fake (a classic borderline delusion). Like Lennon's shooter. He read way way too much into a tiny piece of speech. He may have been stalking her for some time. Lennon's assassin met him earlier that day and tried to communicate too.
Hey listen to this, “Why Can't I Be You?” by The Cure. It's the classic borderline anthem.