If a pickup game of basketball isn't competitive, then does that mean basketball is not a sport? Also, the definition says competes, which means to strive to gain or win something by defeating or establishing superiority over others who are trying to do the same.
Fits chess perfectly, no?
Well, you got 3 out of 5. Even those aren't always the case. Chess isn't always competitive. And it doesn't involve physical skill. And little (sometimes no) physical exertion. So the definition of sport is accurate. If you still disagree with the definition look at the examples given, baseball and soccer.
It doesn't matter whether it is sometimes competitive. Football can be for fun, but it can also be competitive. The same goes with chess. (I'm assuming that's what you meant by competitive.) It only has to involve skill. It can be physical or mental. Notice that the adjective physical is only affecting the word exertion and skill is put separately after the conjunction.
So your position is that it doesn't have to be competitive. We can leave that part out if we like, right? What about entertainment, can we leave that part out too? And skill, what about that part?
If we can pick and choose which parts of the definition we like, or don't like, we can simply say a sport is any activity....... well, ok, just any activity. Right? Can we even leave that part out also?
How about a sport is, literally, every thing. Not just every thing, but all words, both known and unknown. Use your imagination.