Hmmm, I seem to have found the answer to this question. In
Regular and semisimple modules
by Cheatham and Smith in 1976, they call a module regular if every submodule is pure. Regular is of course an overused word, and maybe other people have called this different things. But the justification makes some sense: if I is a 2-sided ideal of R, then R/I is a regular R-module if and only if R/I is a von Neumann regular ring.
