Until recently, I've always considered the phrase "completing the game" as just that - completing the game completely, through the credits to the typical "the end" that many games display. It wasn't until recently when I got into a debate with a couple of friends about game completion and drew to the conclusion that there are two basic categories: "completing the game," and "finishing the story."
The phrase "completing the game" should be used when the player absolutely finds everything imaginable, or at least close to it. This includes any extra side quests, hidden items, completing 100% of everything, etc.
The phrase "finishing the story" is only used when just the basic game completion for the base story is done. In other words, a player breezes through the game with little deviation from the main plot.
I guess I'm more the latter than the former, due to the sheer number of games I own. For the most part, I'd rather progress through the game's original story to its conclusion rather than play a few more hours just to find an extreme powerup that won't make much difference when I need it.
Of course, that's not to say I never try to "complete" any games - it just takes time to get everything, and time's not exactly plentiful at this point of my life. Time is life's currency, and I'd like to spend it more wisely.
Under those circumstances, I guess you can say I'm nickel-and-diming my way through my video games.
How do you guys like to finish your games?