The Punisher has 1) a distinguished symbol (the skull logo) and in the comics for a long time had a costume (the typical spandex/boots/gloves); 2) he fights crime, even though he’s an anti-hero; and 3) a codename.
(Lara Croft does not have a codename, no one actually calls her ‘Tomb Raider’)
As for Drax (and Gamora), they do lack codenames and costumes, but they do have powers (at least compared to normal humans), a heroic mission, and they’re part of a superhero team (which DOES have a name) and fighting alongside other superheroes and against supervillains. So using Busiek’s model (good post, The Other Waldo Pepper) I’d argue they’re 3s.
I’m not familiar with Avatar enough to say, but you might be able to convince me it’s a “fantasy universe” spin on the superhero genre. I think certainly a large part of being able to say “oh that’s a superhero” is the fact that they generally exist in our time period and everyday life. That gets stretched with some characters, like the Guardians of the Galaxy, but they do often (eventually) come to interact with the Earth-bound superheroes.
EDIT: Actually, analyzing Lara Croft through the Busiek model, she fails all five - no powers, no codename, no costume, no heroic mission, and no superhero milieu.