So Wizard is the class, but the title inside of D&D depends on the school the Wizard specializes in. Much like how we call a "Doctor" also a "Brain Surgeon", the "Wizard" could be an "Enchanter", if they specialize upon memory-rewriting / charming / brain-bending magic. (Ex: Charm Person, Suggestion, Geass, Dominate Person).
But IIRC, there is no "Magician", as you said. But there are Evokers, Universalists, Conjurers, Diviners, Illusionists, Transmuters, Enchanters, Necromancers, and more. There are also Sorcerers (_totally_ different), Witches, Magi, Alchemists, Clerics, Druids.
Frankly, I'm personally surprised that there's no "Magician" yet. Its like someone stepped into D&D and surprisingly guessed at the one word that wasn't actually in the game yet... (The closest word, "Magi" is a "Magus" IIRC, not a "Magician")
1. Medium
2. Seer
3. Conjurer
4. Theurgist
5. Thaumaturgist
6. Magician
7. Enchanter
8. Warlock
9. Sorcerer
10. Necromancer
11. Wizard
12+ Wizard (nth level)
...
16. Wizard (16th level or Mage)
17. Wizard (17th level)
18. Wizard (18th level or Arch-Mage)
Screen shot of a scan of the first part of the table (it spans two pages and the 2nd page is uninteresting): https://imgur.com/a/WOS2TAF
I think you're right though. No one with eyes on a big audience is going to want to tie their brand to a Wizards product after this decision. Shame.