Do we call a man who tears a a child out of a burning building a vigilante because he is not a fire fighter and has no powers? No we call him a hero. Hero is night defined by being super, being hero is about the actions you commit as a person, helping others in need, being the person someone else needs you to be. Gotham needed a savior, batman became that savior, Bruce Wayne does not need to fight crime, he does not have to save people, he does because he's a hero.
A vigilante tends to be someone new to the whole hero thing.Vigilantes tend to just be new heroes,without much support,and not really recognised to the public.Then again,I am speaking about how society tends to work in fictional worlds,and not real life situations.
A person doesn't need to have any special abilities to be a hero. They just need to do good and have people that look up to them. A hero is someone we can depend on, super or not.
Refer to the second part of the definition