To the upthread comments, what is the line between hero and vigilante? It is simpler for high-powered "superheroes" who take on world-destroying threats. Nobody gets labeled as a vigilante for taking on Galactus.
But 'street level' heroes are almost all technically vigilantes (excluding any who are actually members of some sanctioned law enforcement agency). Iron Fist, Luke Cage, Daredevil, and similar all take on crime and corruption we normal folks can relate to. We may easily label those who do not kill the "bad guys" as heroes. Like how Spiderman commonly leaves thugs webbed up for the police to arrest.
Frank Castle rides a different line as Punisher. I think you're right it depends on how you feel about our justice system. For example, those crooks Spiderman leaves for the police may be catch-and-released too easily for your taste. Frank's archetype includes the "Old West" justice mentality -- bring 'em in dead or alive, preferably dead, which we do often celebrate culturally. I certainly enjoyed Steve McQueen (Wanted: Dead or Alive) and Richard Boone (Have Gun Will Travel) as bounty hunters in my formative years. Also Chuck Connors (The Rifleman), who was clearly a vigilante most of the time (not usually deputized). But it isn't clear cut since there have been effective criticisms of that behavior more recently as being uncivilized.