Went to a pretty racially mixed elementary school in Georgia in the early 90's. We white girls *definitely* knew Down Down Baby as a story of white aggression:

"I like ice cream
I like tea
I like a white boy and he likes me
So stand back, black boy
You don't shine
I got a white boy to beat ya behind!"

I don't remember ever seeing black girls doing that rhyme, so I don't know if they did it differently. But as a child it made sense to me that the rhyme would assert white dominance. It was just another example of the casual racism we were immersed in in rural Georgia. Even at that age my white friends and I understood that a white boy beating up a black boy for flirting with his girl was the expected norm, not the other way around.