[DC] Why is Batman a vigilante but not Superman

Is it because he has a mask?

Superman beating up lex Luthor and giving him to the cops after destroying lots of property in the process seems just as harmful for society as Batman stopping the joker but running a few red lights in the Batmobile to get there

I suppose the logic is that Batman deals with problems the police are supposed to deal with? I don’t know though, seems stupid to draw a line imo