A linguistic excursion not related to the substance of this controversy: I'm looking for a term for "jokes or puns where the morphemes between which a clever r

A linguistic excursion not related to the substance of this controversy: I'm looking for a term for "jokes or puns where the morphemes between which a clever relationship is drawn are actually true cognates, but the connection is no longer apparent to native speakers". Here that's true because English ad < advertisement < Middle French advertissement < advertir < Latin advertere 'turn towards, notice' is the same ad in "ad nauseam" 'to the point of nausea', both meaning 'toward' in Latin. Does this kind of situation have a well-established name? (There might be subcategories for the case where the cognate preserves a connection in the meaning that's part of the joke, and the case where it's just coincidental, like here.)