Upvote:2
I am not able to give a full answer, but I wanted to offer a partial one, I stumbled upon: My guess would be, that some of this can be traced back to Galen's "The Exercise With The Small Ball"[^1] ("galeni de parvae pilae exercitio"). I think this could be the origin of the ancient medicine ball, because Vivian Nutton, a british historian of medicine, who wrote a book in which he states as regards to Galen's text:
"But if one wished to exercise oneself Galen recommended using a small medicine ball to tone up the body."[^2]
The exercise/sport to which Galen refers to here seems to be Harpastum. This in turn was a romanized version of the ancient greek game Episkyros. I could definitely see how some people would refer to the devices used for those games and exercises as medicine balls.
[^1]: Singer, Peter: "Galen. Selected Works", pp. 299-304
[^2]: Vivian Nutton: "Ancient medicine", p. 242