We had (Glasgow late fifties-early sixties)

eenie-meenie-miney-mo
catch a N----- by the toe
If he cries let him go
eenie-meenie-miney-mo

my mother said that N----- was a bad wrd and that we should use 'darky' kind of hard to believe now. Kids nowadays use 'spider' which works well with the rhyme.

The 'I went to a Chinese restaurant' one was a pat a cake clapping type song, I never heard it used for picking who was 'it'.

We had the 'one potato, two potato' song as well, counted out with closed fists held in front being bumped as opposed to the pointing finger.

fx