I think it may be an American English thing to conflate the two. In English (as a think in Latin also) a professor is a highly learned person - though they may also teach.
You can be a teacher and you can be a professor, but neither as a direct consequence of the other.
Because they want teacher = magister/magistra, and professor = professor.
I guess that professor is higher than magister/magistra, because they teach older children or adults. Like instituteur/maître (primary schools) and professeur (secondary school and universities) in French.
(For French speakers: they created "professeur des écoles" recently, but it's another debate).
Enseignant is the broad term, describing both, instituteur (maître) and professeur, but English languages has only "teacher" to describe this category.
Ludi magister is like the French maître d'école.