Hmmm, I seem to have found the answer to this question.  In

[Regular and semisimple modules][1]

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.  

  [1]: https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-65/issue-2/Regular-and-semisimple-modules/pjm/1102866791.full