[aidenn0]

Jimmy is a diminutive form of James. It's the sort of thing where parents would call a kid "jimmy" and when they grow up they will ask people to call them "Jim" or "James."

Adding "little" in front just stresses the fact that we are talking about a kid, so the references are about dumbing it down for kids.

Johnny is probably a bit more common than Jimmy, but I've seen both. See e.g. [1]

1: https://en.wikipedia.org/wiki/Why_Johnny_Can%27t_Read