I'm not aware that 'do/play' refer more to BE or AE, or vice versa. "do" and "play" are used for sports and activities. When we use "do" or "play" depends on the sport or activity. These might help you, though I'm not certain that it works for EVERY situation. "Play" is used with ball sports or competitive games where we play against another person. e.g. "We play tennis every week." But, we don't "play karate" "Do" is used for a recreational activity or a non-team sport that does not use a ball. Think of "do karate" So we can "do sports" and "play sports"!

I do not know what GE or SE stand for. I think Helen plays sports. In fact, she does a few sports. John does no sport whatsoever. He doesn't play any sports. (It sounds a bit more natural to put a 'few', 'some', 'many'. They are not absolutely necessary but when they are added in, it seems to flow better!)