With me, the deffinition can pretty much be summed up like this.
If the game isn't fun to play, then it doesnt matter how good the characters, story, or graphics are, because you won't be able to enjoy it.
I've played several games that had good stories, characters, enviornments, messages, etc. but they were boring as hell to play, and ultimately I hated them because of it. What bugs me is when some people say "games are an artform, and everything else is secondary", and believe that making a "fun" game is simply pandering to all the simpletons.
Games can easily have both, and when the two are combined, they are what make the best games. Bioshock had a beautiful enviornment, a cool story, and most importantly, WAS FUN. By combining all these aspects, it made the game better in every way, and helped draw me in more than if they had been extremely serious and said "Stop having fun guys, we have a story to tell you."
I think its what a lot of people mean on the forums when they say they want more "fun games". We dont want serious stories, great characters, and all that hard work to go away, we just dont want to see it continue at the expense of making the games dull and boring. As an example, it's part of the reason games like Bulletstorm, Duke Nukem, and Serious Sam are getting so much hype at the moment. The FPS genre has made big steps in presenting more serious storylines and characters, but it feels like the advance came at the expense of making the games feel like cookie cutter copies of each other (people love to call out COD style games, but think about all the Halo and GoW knockoffs too)
TL;DR I would rather have a fun game with a shitty story over a well written game with dull gameplay any day. Mature gaming experiences with great stories and characters are important, but it's all for naught if the games stop being fun