When I searched for what I remembered as the "original version" I found references to the "Shortnin' Bread" version, though with even more of a racist tone than the one you posted (instead of "5 little monkeys", it has "5 little d*rkies" and the doctor's advice is to feed them shortnin' bread instead of not to jump on the bed). I couldn't find any references to the version with "one fell off and broke his head." Now, searching specifically for that line, I found a couple versions with that line in it, so I'm not losing my mind and I did remember that version from my childhood. Thanks.