Professions by google image search:
In the games industry, being a great graphic designer will make you suitable to work as part of the user-interface team.
Being a great illustrator or painter will make you suitable to work as a concept artist or texture artist.
3D modelling skills will make you suitable to work as an environment, prop or character artist.
Often, game artists will be skilled in multiple of these areas so that for example, they can do sketches of a character's anatomy (drawing on their classical education), make a rough model of it (using 3D modelling), do a "paint over" concept (using digital painting), refine the 3D modelling, and then paint the textures.
If you want to make the art in the actual game itself, then a classic art degree and/or a multimedia art degree will be more useful than a graphic design degree.