Any thesaurus can provide you with terms to refer to people who can work magic. Wizard, witch, warlock, sorcerer, enchanter, magician, mage, magus, charmer, necromancer, thaumaturge, thaumaturgist, occultist, alchemist, conjuror, theurgist, and more.
Some works of fiction -- even works of fantasy -- have used them as a form of elegant variation to refer to the same character. This is unwise.
And not merely because some of those have additional connotations. "-mancy" as a suffix is no longer really connected to the notion of divination, but "necro-" still means death. You do not have to have your necromancer summon ghosts to question them about how to find buried treasures, but if he throws fireballs rather than orders skeleton armies into battle, readers will wonder. Likewise, an alchemist should pother around with bottles and crucibles and probably discuss turning lead into gold.
Some are more obscure, but the very way that theurgist is likely to be unknown will lead some readers to look it up and discover that a theurgist, in the Roman Empire, would claim to work magic with the gods -- not like a D&D cleric, they claimed the power to compel the gods, and also they claimed that only very pure and good people could practice theurgy, and that most people who tried it would get daemons instead -- and they were still prosecuted for impiety by both Greek and Roman pagans.
Despite all this baggage, it was sometimes known in pulp eras for writers to use the term merely to mean a practitioner of white magic, but that usage is very rare nowadays, and for good reason.
(Not to mention that exotic and far-fetched terms need work to make them seem as familiar as they are to a viewpoint character. If theurgists come from a far-off land, that's one thing, but if theurgists work next door to the bakery and buy a lot of pastry, that's quite another.)
But what about the more generic terms? Wizard and sorcerer have been thrown around so often that the specific uses have been rubbed off. So you can use them interchangeably?
This is a very foolish notion if you have more than one type of spell-slinger. Each one should have its own name, and it should be used for characters of that type only, and their natures clearly established at least according to the viewpoint character's knowledge, so that the readers can tell them apart, and know which one is which.
Or not, if the characters in the story lie. But to discover that an apparent wizard is really a sorcerer means nothing if you have been interchangeably referring to wizards as sorcerers, or sorceresses, and vice versa.
This applies all the more as the differences increase. If a wizard intones a spell while an enchanter sings it, that's a difference. If a herbwife knows how to find the white flower that grows only about springs that unicorns purify, and can cure poisoning while the venefica knows how to conjure deadly poisons from thin air, that's a large difference. If a wizard compounds potions while a sorcerer conjures devils and sets them on his enemies, that's a major difference.
Then, what if there is only one kind of magic and therefore only one sort of wielder? Or, at least, it's more logical to regard the types as subtypes, and refer to, say, earth wizards and air wizards, or as the case may be, water sorcerers and fire sorcerers? It would be fine to use such terms then, wouldn't it?
No, it would not. Consistency in usage is one way to build up the world-building, and make it clear that there is only one type of person wielding magic in this world. It's one thing for the knight to have his sword be sometimes a blade and sometimes a weapon, or his mount be sometimes a horse and sometimes a steed -- if the style calls for it -- but those things are known to be a sword and a horse. The unity of wizardry has to be demonstrated.
I call all my people with physical magic "wizards" though my exception to it is that I have a category called "battle mage" though only as a subset. Seers are a type of wizard (according to most other wizards), though typical people do not call them that (or think of them as such) - it is more of a cultural thing. My reasoning is that all "wizards" inherit their magic, but the specific magic ability is based on their experiences/education that cause them to develop in their specific ability. The simplicity of this magic system makes it easier keep track of what people are while allowing a great diversity of abilities.
The inheritance part though, is very complex, but that I leave for a long article.
But the naming remains rigid, not variable, in order to avoid confusion - which I agree is necessary to make the story easier for the reader.
I skimmed down to your final paragraph. Have to agree with that premise. We can get too carried away with our vocabulary. We're not writing thesauri or word-a-day lessons. We're writing fiction. And a good writer can make it work.