The dictionary definition of sport is not the definitive authority when it comes to parsing what qualifies and what doesn't. The superficial notion that a sport must involve a certain amount of physical activity is insufficient. Who's to say how much it takes, must you break a sweat first? That is quite a loose argument.
The real differences between a game and a sport aren't so obvious as to which involves more bodily movement. No, the real difference is whether or not the game within it's rules is limited i.e finite in expression. Checkers has been solved, it is limited and is therefore relegated to merely a game. Chess is near infinite and is for the moment a sport, the 64 squares is enough.
It can be argued quite well that everything is finite. Furthermore, are you saying that chess is a sport now but will cease to be if it ever gets solved? That's sorta ridiculous.
Well what happens during a basketball game isn't "finite" and everyone agrees that its a sport. The number of possible chess games is for all intents and purposes infinite, and that doesn't even matter considering for the context of two people playing a game there is no point at which the ideas will run out. There is no sense in pointing to the abstract and saying something like "well if a supercomputer comes along and solves chess"-then it would be a merely a "game" to the supercomputer because an end has been reached. To us humans that point is moot. The possibilities for us wont ever be exhausted.
How about this: if you can tell someone what to do and just have them do it for you, it isn't physical enough to be a sport. You can do this with chess, you can do this with checkers, you can do this with chess, you can't with pool or snooker or golf. Of course that only takes care of a physicality argument, and there's still other issues of being a sport - i.e. rope-climbing is very difficult physically, but it isn't a sport.
Note: I just thought all this up, so it may not be foolproof, but it's a stab.