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.
He would not be a superhero. Superheros have empathy and realise that doing a crime does not immediately make you a bad person worth killing.
Do you really think that a superhero would kill a dad for robbing a service station to pay for their kids medicine?
Look who I am talking to, of cause you would.
What you are writing is an antihero like Punisher or Deadpool