I tend to think of a hero as a cultural expression.
For example, Superman is a hero. In his last film, he managed to cause enough collateral damage to wipe out a town. Some people complained but the average movie-goer seemed not to care a whit; it did not clash with their concept of heroism.
Those who did complained are even more interesting though because their complaint only makes sense if being a hero can exist independently of performing heroic actions. If they already *know* that Superman is the hero because of popular culture and the way the story treats him, and they're reacting to someone they *already see* as a hero acting the wrong way.
If you don't presume Superman is a hero, there's nothing to object to when he slams another character through a store we've seen people just flee into. The Hulk does this frequently, but we don't presume a screaming green mountain of rage is a hero, so no one suggests that it's inappropriate for that character to perform that action.

This becomes more explicit when we look at red kryptonite. It causes Superman to 'turn evil' by radiating his cells in an odd way, as though Superman's hero status is packed into his mitochondria. Whatever bad behaviors he engages in is a physical reaction to a noxious stimuli - like a tend to get sinus infections in late spring.
Previous editions of Dungeons and Dragons (and so a tone of fantasy literature) also described heroism this way. You could sniff out how much heroism a person had in them in 3.x and, if so inclined, stuff them with villainousness like some sort of demented housewife might pack a Thanksgiving turkey.
And I'm... drifting here. I think I'll quit for this post.