Every little kid I knew used this rhyme frequently and we always said “tiger.”
As someone who was a kid later than that, it was always “tiger.” It’s how our mom taught it to us when we were little and how we always said it to determine who was “It” when we were playing hide & seek and other kids’ games or to decide who got the last cupcake.
The rhyme really only makes sense with the original word, since tigers rarely carry cash.
A lot of kids’ rhymes don’t make much sense. Fortunately, we never worried much about things like that. 😏