I keep forgetting to check on this when I come to it, but some professions do have an article (a/the) and some such as doctor and journalist do not. I may be misremembering it (and unfortunately can't remember which lessons these are in to go look for the specific examples), but it seems like there are some professions mentioned which do use an article such as professor.
Maybe it's that the conversations with professor are speaking about the professor, but I'm not sure. Professor and teacher is spoken of in a different section from where I've encountered doctor & journalist, I think, so it's hard to compare directly because they aren't popping up at the same time.
It's confusing with the professions, if they are as you say, because words like son and daughter do have the definite and indefinite articles with them.
This is maybe the trickiest thing to me, what seems to be inconsistency with gender and definite/indefinite article and preposition for in or by, etc. It may be that (like with English) there are just some things to remember the way they are, and that there isn't a rule for it that works in every situation?
“Medician” is not a word in English. And a doctor can indeed be a profession, as a physician is a medical doctor. Or someone with a PhD is also a doctor - who has a doctorate in any number of fields. So the word, “doctor” can refer to both a profession, such as a surgeon, and a doctoral level academic. Sometimes that doctor can be a professional and an academic - as for example a psychologist who has a doctorate.