Not quite. I would say that a professor is a teacher at a higher college university, which would includes things like 2 year community colleges where a PhD is often desired, but not required. It does however suggest a level of learning beyond the "teacher" designation that has been earned through additional studying and/or experience beyond a bachelor degree.
English sometimes omits the article too in sentences like this, most notably "I am king". However in this case I'm wondering if it's not like some other languages (Spanish etc) where you omit the article when describing your profession because it's considered an intrinsic part of yourself. So like "I am strong", "I am tall", "I am English", you'd say "I am professor". Unless of course you are saying that you just one of a larger group, "I am a professor at the university".