Game developer? Yes.
IMO you should only get a degree in game development to enlarge your chances for contract work.
If you're your own boss and you have years of experience however, I don't see the point of getting a degree.
In fact, I tried getting a degree for game programming, in which I failed because I'm much better in actually doing it rather than showing what I know about it on paper.
I can easily prove I'm an excellent programmer and that I can learn new stuff very easily when I programme what I have to do.
But if I need to answer questions on paper to show what I know, I'm always doomed.
And since most exams were multiple choice questions, you can perfectly see why I failed.
Long story short: you don't need any degree to claim the "game developer" title.
When developing for PC or mobile, you most likely don't even need to be a company.