I loved reading this!! It brought back a lot of memories. I remember another jump-rope song I used to play in Kindergarten (1976) :

Down in the valley where the green grass grows,
There sat (girl’s name), as sweet as a rose,
Along came (boy’s name) and kissed her on the
cheek,
How many kisses did she get that week ? and count as you jump.

Also, we played a jump-rope game called “School”.
You ran in and jumped once and ran out for 1st grade, ran in and jumped twice for 2nd grade, etc.