User blog comment:TheKing44/Grand Finale Functions/@comment-35470197-20190611051239/@comment-35470197-20190611070818

They are actually called first order set theories even though you are not regarding them as set theories. You might think that they should be called as "set and class theory" or something like that, there is no such terminology. "Set theory" does not mean a theory with \(\in\)-relation which is compatible with axiom of ZFC. For your context, ZF+AD is not a set theory because it conflicts ZFC? It is not a standard terminology.