Heros and anti-heros are united in that they genuinely want to do good. They just differ in the means (or what they'd allow to use as means) to their ends. For example, a hero will vehemently refuse to blow up a street market while an anti-hero might consider it if they deem it to be sufficiently helpful to their end.

I'd rather look at it as a sliding scale with “will never do anything bad” on one end, and “a villain who has good intentions” on the other. And even those two ends are subject to the questions “What do you mean by ‘’bad?” and “What do you mean by ‘good intentions’?” Thus, I think while heros and anti-heros across stories and genres have commonalities, one story's anti-hero might as well be a hero in another story and that the best way to judge a character being a hero or an anti-hero is in the light of the story they're in.