So they're not called wizards because that evokes specific images. Gandalf, Merlin, Harry Potter. They're not called witches because that evokes something pretty specific too. Margaret Hamilton. Snow White bitch. Ron Weasley. They're not sorcerers or mages or spellcasters or enchanters or illusionists or bards or mystics. They're Magic-Users.
There aren't always knights and paladins and rangers and mercenaries in old school D&D but there are always Fighters. Thief is a pretty specific appellation but they also cover pirates, assassins, ninja, and so on. We have Clerics instead off priests, prophets, friars, Benedictine monks, or Jedi. Each of these very broad classifications can cover any of these kinds of characters or ones I haven't listed. Not just archers but like pikemen, not just pickpockets but like spies. So on. They're just grouped according to what they use to get the job of exploring a Dungeon or fighting a Dragon accomplished, but also imply their use to a specific end. Fighters use weapons and armor to help them fight. Clerics use their faith to help them survive and care for others. Thieves use skill and guile and surprise to steal things. Magic-Users use magic.
A Magic-User is a Fighter who decided making people pass out so you can stab them in the eye was a good way to fight. A Magic-User is a Thief who decided that Knock would be more generally useful than Open Lock. A Magic-User is a Cleric who decided that the gods who have temples are great and all but The God In The Bowl is pretty righteous, too.
That's it. I mean there's a mystique to it but the default D&D assumption is that anybody can use magic with enough time, study, and fortune. You have to sacrifice some things (you don't have time to practice enough to get skilled at other more esoteric practices, will never be as fit or hardy as a fighter, you've probably pissed off the gods too much to ever get much out of them) but you get a wide range of abilities and some very powerful ones. The Magic-User spell lists would befit a politician as much as they'd be useful for slaying a manticore. And that's entirely possible because the only end that the existence of Magic-Users as a class implies is existence as a Magic-User. That knowledge and growing more powerful is its own reward.
Lamentations of the Flame Princess implies that all Magic-Users are twisted and shunned from polite society, which is why there are so many adventuring Magic-Users. I think that's a good idea but I think it's as likely that the kind of people who don't fit into society anyway and go a-wandering are the kind of people who put in the trouble and struggle to learn how to use magic in the first place to get some more specific end. I think that interpretation sits well with more people than even realize it because it's a specific end and the method by which it's accomplished that would really differentiate someone's approach to how they use magic and for what and would obviously produce some vastly different Magic-Users in the course of play, from a pure role-playing perspective.
1) If a normal regular Magic-User is not to your liking, you can play one of the mere specific/weirder magic classes people have introduced over the years, from Illusionist to Muscle Wizard to Bard. Know of course that you're usually still dealing with a very Vancin structure, if not Vancian methods. However before you make that decision know that....
2) There's nothing stopping you from playing a normal Magic-User and us changing some things very simply and keeping you very the same while feeling very different.
- You want to be a bard? You get a free instrument, I cherry-pick you bardy spells, and you can cast magic by playing music. You can use your instrument as a weapon, El Kabong/El Mariachi style, but if it gets broken as a weapon it's broken as an instrument, too, and you can't cast spells til it's replaced.
- You want to be a sorcerer? You don't have to keep a spellbook, but you as a player also don't get to write down all the spells you know; you have to remember them or someone at the table has to remind you, and you're still stuck with only so many slots per spell level.
- You want to be a Magus or Swordmage? Normal progression of spell slots, slots are "charges" you can expend for free in a round to give yourself an AC bonus equal to the spell level or do extra damage, or they may be filled as a spell normally with all touch spells or spells targeting the caster being cast on the sword.
- Enchanter, artificer, magical armsman? Normal spells and you can wear chainmail or less but can't use shields, you can NOT cast spells but instead store them in weapons, armor, and items for one day, and you (or whoever) trigger them using a touch and a command word.
- Alchemist? Spell slots are potions you can brew per day. A potion of that spell level can be anything from the normal spell chart for that level but suffers a percentage of Failure and a much higher percentage of Side Effect.
- And so forth. There's lots of little changes we can make. Hell, you want greater spell flexibility or more spells per day? We can talk about that, tradeoff and consequences, like DCC spell failure and backlash effects or something.
That's what D&D is about, really, at least these old school versions like I'm running: you want something special, we can talk and bolt it on no problem, and we don't need Mike Mearls' seal of approval to do it. Let me be clear I have no problem with all the different class variants, but say the Scrivener's Enchantment I did? There's no reason I can't just say "Have an Int of 9 and you can cast a spell or do that for each slot." Having that codified and written out for people to browse and take and use as an off-the-shelf option is cool, sure, but it's not necessary to make a whole new class and new subsystems and rules to the extent of full class design for every little change from Strict Vance. But neither does changing something about the Vance style (full range of movement, some physical components, magic words, memorization that's like putting a demon in your head and it's hard to do anything else) and D&D spell structure (spell levels, so many slots per spell level per character level) aren't necessarily married til death do they part, and a lot of what gets accomplished by reinventing the wheel altogether can be taken care of by just fiddling with one aspect or the other quite slightly.