1) John is the professor of English at ABC University.
2) John is a professor of English at ABC University.
Does the first sentence mean that there is only one English professor at ABC University and John is the one? Does the second sentence imply that there are more than one professor at ABC University and John is one of them? And what if we omitted the article altogether? What would that mean?
Yes, ‘the professor’ suggests exclusivity and ‘a professor’ suggests one of others. If there is no article, you would simply be describing John’s occupation and not indicating whether he is the only one or one of a group.
might also imply that John is the most noted professor (for competence, queerness, etc.), I believe. That, of course, is in case you put the stress on “the” while speaking.