"Shooter" feels like a fast and easy term to me. "To heck with the permits, just get me a shooter and some talent!!" To me it implies one-man-band type low budget productions.
I find I use Director of Photography and Cinematographer almost interchangeably. Although, I'd say a Cinematographer is more likely to be operating the camera (in the context of a big budget film).
When I was in film school there was great confusion over the slate. The slate clearly read "cameraman," to which we wondered "Why should we put the name of the camera operator on the slate, and not the Director of Photography?" After a couple of amusing blunders we were corrected. We were told that "cameraman" is a hold over from the early days of film when you had some a actors, a director, and that other guy that made the movie. A term which stuck, apparently, until recently.
Whatever name we pick, it's a marketing tool and should reflect our experience and job aspirations. And like any marketing tool it is only means what the people who are hiring us think it means.