For my whole life, I've been asking sports people to define "sport" (as opposed to a "game"). Nobody has ever been able to provide a good definition -- this appears to be a "you know it when you see it" thing (and I apparently lack the ability to know it when I see it!)

Same. I've asked several workers in the field over the years and have found it to be based on two things: first, on several generally overlapping characteristics, all ranked more on spectra than as binary requirements, and second, on local culture.

An example of the first is that shooting is a little bit physical, so it often checks off the "physically strenuous" box (if barely, but still more so than chess). An example of the second is ex-Soviet Bloc states focusing on strenuousness and competition with others, Scandinavians on health, well-being, and more so competition with oneself, and North Americans on trying to nail down just how physical the activity is, and if an audience is present. These are just off-the-cuff conversations over about 2-3 years of work in sport but still interesting to think about. Some have been recorded on https://spmx.ca/tag/sport-psych.