In high school when I was asked whether I played any sports or not, I would often reply "no." Then quickly add the caveat that "play" was not the right verb to describe my participation in Nordic skiing and running races. They're certainly forms of sport, since they involve competition and physical prowess. But, they aren't "played" in the sense that they don't require a sequence of moves to win or be competitive.
Alternatively, sports such as football or tennis are sports which are "played." I would suggest that using the verb "to play" implies that the activity requires the enactment of a strategy involving sequential moves. For example, setting up a point in tennis (against a competent opponent) requires that you opponent not be able to return one of your hits. This can be achieved by the successful execution of a combination of moves. Having some physical prowess is necessary to execute a sequence of moves against a competent opponent in Tennis, but pure physical prowess won't win any games. Because of its reliance on strategy to win the competition, Tennis (among other activities which are "played") must also be considered a game.
Finally, when considering Chess, the execution of a sequence of moves is critical to winning against an opponent. Physical prowess is not important to the execution of a winning strategy in Chess. Because Chess does not require physical prowess but does require the execution of a sequence of moves to win, Chess is properly considered a game.
To summarize classifications:
I think a running race is properly classified as "Sport" since it relies almost entirely on physical prowess. Tennis is properly classified as both "Sport" and "Game" since it requires both physical prowess and strategy, while Chess is properly classified as "Game" since it requires strategy but no physical prowess. I don't intend to demean Chess by not considering it as a sport. It just doesn't require athleticism, which is characteristic of nearly all activities which are considered "sport" here in the United States.
I may be biased anyway since I always preferred running to playing chess. 😉