My Mum used this position for enemas when I was growing up, and I am not quite sure why - it has little to recommend it, certainly if the comfort and the dignity of the recipient is considered. It gives maximum exposure of anus and genitala, with a sense of total vulnerability and exposure. It seems to me that Simms is the best all round answer, given the trade off between dignity and efficiency.
Her enemas were given with a Higginson, and were always a pint to a pint and a half of soapy solution made with Lux Flakes. I had to keep my legs up myself and opened wide to each side. I was towards the edge of the bed, with the solution in a glass bowl on a tray. Bowl, Vaseline and syringe would be brought into my room. Enemad were never given in the bath room.
Today my spouse and I use the Simms position. I am sure on all fours bum in the air is best for effectiveness, since gravity helps the flow up the colon. It is also easy for nozzle insertion. However, like on the back legs up, it also has a greater exposure, which might bother a child more than an adult!
I am not sure of the origin of the on the back position, since I am sure it is not used medically - at least not for an enema, or the real benefits it offers. I am also not sure how common it is or was in the USA for giving an enema to a child in what is called the Golden Age.