Then we get into the conflict of terms when we refer to a 'baseball game' and a game of 'basketball', which have formulaic fields of play, yet by most anyone's standards are sports played by athletes. Maybe each individual competition can be called a game, and the institution is the sport. All of which I think we explored in past threads, with circuitous results.
I read Behr's explanation and was very much influenced by his thinking.
Your note about basketball and baseball being sports is clearly the modern usage, but I still like his distinction.
For the most part, games include:
1) The requirement of an opponent. (You can't play baseball by yourself.)
2) Defensive play, or play upon the opponent. (You can physically prevent your opponent from scoring)
3) Both "teams" use the same ball.
Games include football, handball basketball, tennis, hockey, racquetball, soccer, rugby, polo, etc.
Sports, on the other hand, generally include:
1)No play directly upon the opponent.
2) No shared ball
3) The ability to play without an opponent, in competition against a standard or your own previsou performances
Sports include darts, shooting, running, horseback riding, golf, swimming, all the ski sports--including alpine, nordic, shot put, discus, javeline, archery, automobile racing (except, perhaps NASCAR <grin>), etc. Biathlon combine two sports, and triathlon... well, you know.
Of course, like all such delineations, there are problematic examples. A no-ball game/sport, like fencing, is a perfect example.
Nevertheless, the distinction is useful to me because there's a different mindset in sports than there is in games.