We did clapping games when I was in primary, but the words weren't exactly like I've seen them. One was 'A-B-C Together' and we did 'Miss Suzy.' I think I've heard the one about the Chinese Restaurant (variant with Elvis Presley).
But this is funny: I also know Eenie Meenie Sicileeny, with different words to the ones I've found on the net. I know it from a two-week holiday in Greece when I was eleven. It was a resort, most of the other girls were English. So these rhymes are still getting passed around! (I never saw any boys do clapping rhymes or skipping rope games - and we never played skipping rope games either).
I think I'm about equivalent of eighth or ninth grade, but I don't know if younger pupils still play these games because there's a split between primary and secondary - when I came back from that holiday, I never bothered to teach any of my friends Eenie Meenie because we were in High School and too 'grown up.'
By the way, Azizi: I live in Scotland. Children still play creative playground games, but they more involve running around - tag, piggy-in-the-middle, Granny Greensteps.