Luigi is called a ‘hero’ because he killed a bad person, while Frank Castle and Dexter both kill bad people (literally all they do is kill bad people), yet no one calls them ‘heroes.
And if I write a “superhero” story about a guy with superpowers and he kills every single criminal he meets, from the bank robber to the crime lord, and leaves no criminal alive, he should also be considered a “hero”, more of a “hero” than Superman.
the archetype you’re looking for is “Anti-hero”. a kind of hero with grey morals and questionable ethics - but whose actions still have a mostly positive result
typically - a hero will try to rehabilitate a criminal first, throw them in prison second, and only kill if they’ve exhausted all other options. killing is usually shown as something that the hero struggles with greatly
an anti-hero will often skip the “when will murder be my only option?” ethical dilemma regular heroes have and go straight to the killing. in their stories casual killing is either shown as normal, or serves as a plot point that’ll come back to haunt them later (“i killed that man without thinking, but now he turned out to be innocent”)
this is not the only difference between the two of course, but i don’t want to type an essay
the punisher is kinda badass
You know who else idolizes the Punisher? Cops.
Cops also like doughnuts, thats not gonna stop me from loving them.