Alvin : It is true that "professeur" can be used as well to include school teachers and lecturers. Notice I said "usually" ,and I meant it is rather used for university lecturers.
It is common to say for example:
Philippe est professeur d'histoire. Philippe is a history teacher.
Your interpretation is correct.
"Professeur" is rather a title, (titulaire d'une chaire probablement universitaire).
while "enseignant" is emphasizing on the job description,namely "teaching".
However it is not common to call a university lecturer"enseignant" ,although he is actually teaching.
Synonyms for "enseignant" according to the dictionary "Dictionnaire Francais Definition Synonyme" in french are 'instituteur, maitre, precepteur '.
precepteur is a tutor
instituteur is a primary or elementary school teacher and maitre' again is a primary school teacher.
Synonyms for "professeur"
'maitre, precepteur, enseignant, precepteur, prof, instituteur, magister, educateur, instructeur.'
* magister means professor , lecturer as a title .
* instituteur is again a primary or elementary school teacher.
Although both words could be synonyms according to the dictionary, yet the distinction is in the act of applying both.