Good point. English has both actor (=someone who acts) and agent (=someone who acts) from agere = to act, and both doctor (=someone who teaches) and docent (=someone who teaches) from docere = to teach.
that's the root of the word. The original four-wise distinction seems to have been Doctor of Medicine/Law/Natural Phylosophy/Theology.
Just different kinds of knowledgeable people.
Interestingly enough I just realized in Italian there is an old word, "dotto" that literally would mean "someone who has been taught" but concretely is used in the sense of well educated, knowledgeable, wise, cultured.
Bologna, the home of the first university, has the nickname "la Dotta", i.e. the educated one.