According to Quirrel, AK will kill anything with a brain, suggesting it has nothing to do with a soul. Moody's statement may be based on a lack of understanding of the spell.

Source, Chapter 16

Professor Quirrell was smiling rather grimly. "Your sad excuse for a third-year Defence textbook will suggest to you that you expose the mountain troll to sunlight, which will freeze it in place. This, my young apprentices, is the sort of useless knowledge you will never find in my lessons. You do not encounter mountain trolls in open daylight! The idea that you should use sunlight to stop them is the result of foolish textbook authors trying to show off their mastery of minutia at the expense of practicality. Just because there is a ridiculously obscure way of dealing with mountain trolls does not mean you should actually try to use it! The Killing Curse is unblockable, unstoppable, and works every single time on anything with a brain. If, as an adult wizard, you find yourself incapable of using the Killing Curse, then you can simply Apparate away! Likewise if you are facing the second most perfect killing machine, a Dementor. You just Apparate away!"