The story
From regular (“continuous”) + black hole (“region enclosed by an event horizon”). From having a spacetime formulation that is without discontinuities, that is, regular.
Etymology from Wiktionary, CC BY-SA 4.0.
regular black hole
Today it means: A mathematical formulation for a black hole which does not contain a singularity. The environment outside the event horizon is the same as other formulations for black holes. Full definition →