"J'ai un rendez-vous avec le docteur."
Translation:I have an appointment with the doctor.
May I correct your French, first?
"en classe et DANS mes livres,.... Sont-ils synonymes, ou est-ce que le registre..; les différences sont-ELLES régionales ?"
Now the answer: the more common word is "le médecin", that's true. But older people tend to say "le docteur". In French culture "un docteur" is always a physician, because the other "docteur" titles (all doctorate tenants, in any disciplin like law, chemistry or else) are not used.
When you talk to a doctor, in French, you say "Docteur, j'ai mal au bras".
My translation, "I have a rendezvous with the doctor." was considered wrong. Although the "correct" translation is fine, I do not think my answer is wrong. In American English a "rendezvous" is a meeting or appointment, especially the former. So, I think my answer could be considered correct.