Your definition is mistaken, though the distinction you're drawing is real.

A sport is a type of game. Every sport is a game but not every game is a sport. There are other types of games that are not sports, and do not share the same quality as sports do.

It's a lot like niggers. All niggers are monke, but not all monke are niggers. Other types don't talk like niggers, they don't commit the overwhelming majority of crimes, but niggers are still monke