I think you're viewing things from a LotR perspective where "classes" don't quite exist, just people who are good at different things. But in that world, it was pretty much uncommon for people to achieve heights of skill that set them far above normal folks (I think) compared to D&D where PCs do exactly that regularly.
So a Fighter is not just a guy who hits people with sticks and blades, he's a highly trained expert with weapons and keeping his enemies away from his allies. He's not a corner cop, he's James Bond.
Adventurers would want to get across as much information as possible without literally describing every single thing they know how to do. Labels help in that way.
An adventuring party is a team, not a club. If you were starting a baseball team and had a bunch of people to choose from, what would be more useful to you as the organizer? "Yeah I think I'm pretty good at hitting the ball. Probably there are others who are better. But I'm really better at catching," or "I'm an outfielder."
"Warrior" is too vague, because anyone who fights is a warrior. "Fighter" in common language is too vague, because everyone who fights is a fighter. But pretending in the D&D world that the word "Fighter" connotes the Fighter Class, it actually carries more weight.
It may help to replace in your mind one word with others though. We as players find it to make perfect sense to describe someone by their class name, but if you think the characters wouldn't, substitute whatever would sound better in your own mind. "I'm a Cleric" could be translated as "I'm a devoted follower of my god, gifted with magical powers that heal and protect, and I can wear armor and use weapons."
Personally I think the "Adventurer's Guild" idea of people applying labels to themselves and others to facilitate "pick up groups" is simplest in the game world, because most game worlds have archetypes of that kind of person throughout their history. Instead of "You've heard of Merlin right? I'm totally like that guy, except not as good yet," they just say "I study the Wizardly arts."
EDIT: People label themselves and others all the time. "I'm a Banker," "I'm a Lawyer," "He's a Teacher," "He's German," "She's a Rhodes Scholar," "She's a bassist," "That guy is an amateur wrestler."
Miliatary, law enforcement etc. all do it as well.
In my play groups, rangers are always referred to as rangers, and wizards, barbarians, sorcerers, druids and rogues are always referred to in class terms with nobody objecting to the meta-gaming. "Fighter" is usually not used, even if the character in question is a fighter. Instead you'll hear "meat-shield" or "tank" more frequently. Paladins are also commonly referred to in class terms.
In fact, the more I think about it, virtually all of the classes are referred to in explicit class terms except the fighter. Even warlocks are casually referred to was warlocks. I think the issue here is just the use of the term "fighter" to describe a class. The word probably is too generic and that causes the confusion.
Heck, my druid, ranger and barbarian are quite likely to refer to themselves as their class without thinking twice about it.
So, fighter alone is a bit off. Its just someone who fighters. Fighter-for-hire means a mercenary. A bit vague, so it could mean anyone who fights for money. But at least workable, and maybe the fighter doesn't feel like specifying because she has a giant sword strapped to her back.
My PCs tend to refer to people by one of several professions, such as refering to all arcane users (Except sword mage) as a mage. Some might be mages who study, sure. Maybe call them book-mages. Others might be pact-mages, or blood-mages or what have you.
Other times, my PCs will refer to them or others using class names as organizations. People who sneak around the forest are Rangers, even if they have rogue class levels. People who live in barbarian tribes are barbarians, even if they are wizards.
Confuses the heck out of other players, however.
Some might be mages who study, sure. Maybe call them book-mages. Others might be pact-mages, or blood-mages or what have you.
Or, "wizards" and "warlocks," and save yourself the compound words.
I think any organization of practitioners of particular styles would want to differentiate themselves from "others" and "inferiors." It's a common trope that wizards view themselves as "better" than other spellcasters. A Wizard guild would probably take great pains to make sure people didn't confuse a Wizard with an anything else.
"Did you just call me a sorcerer? Sorcerer!? Don't lump me together with those brainless hacks, who think they're so great just because they can make sparks fly out of their fingers naturally! I understand the laws of magic, and bend it to my will. I have studied long and hard to master the secrets of the universe! My knowledge far exceeds some mutant hedge-mage who can spit lightning out of his butt when he sneezes!"
Any such indignation on the part of Wizards would then be taken up by others, possibly in response to the Wizards: "Don't call me a wizard! Those losers cover up their lack of talent by wasting their lives away in dark libraries! They can only come close to imitating the natural power that flows in my veins! No, I'm no wizard, I'm a Sorcerer!"
"You think I'm a Warlock? I haven't sold my soul to some devil-man for a trickle of undeserved power! Hell no, I've trained for years mastering the interplay of swordsmanship and battle magic, following the great tradition of the Eladrin masters of the Coronal Guard. I'm a Swordmage damnit."
That having been said, if you have anything resembling an "Adventurers' Guild" it's ok to assume that all those specialized fighters mentioned above could easily be lumped under the heading "fighter" by the guild's leaders.
Some might be mages who study, sure. Maybe call them book-mages. Others might be pact-mages, or blood-mages or what have you.
Or, "wizards" and "warlocks," and save yourself the compound words.
I think any organization of practitioners of particular styles would want to differentiate themselves from "others" and "inferiors." It's a common trope that wizards view themselves as "better" than other spellcasters. A Wizard guild would probably take great pains to make sure people didn't confuse a Wizard with an anything else.
"Did you just call me a sorcerer? Sorcerer!? Don't lump me together with those brainless hacks, who think they're so great just because they can make sparks fly out of their fingers naturally! I understand the laws of magic, and bend it to my will. I have studied long and hard to master the secrets of the universe! My knowledge far exceeds some mutant hedge-mage who can spit lightning out of his butt when he sneezes!"
Any such indignation on the part of Wizards would then be taken up by others, possibly in response to the Wizards: "Don't call me a wizard! Those losers cover up their lack of talent by wasting their lives away in dark libraries! They can only come close to imitating the natural power that flows in my veins! No, I'm no wizard, I'm a Sorcerer!"
"You think I'm a Warlock? I haven't sold my soul to some devil-man for a trickle of undeserved power! Hell no, I've trained for years mastering the interplay of swordsmanship and battle magic, following the great tradition of the Eladrin masters of the Coronal Guard. I'm a Swordmage damnit."
This works fine if the character subscribes to the standard fluff. You can most assuredly have, say, a sorcerer who's learned his craft carefully and methodically, or a warlock who doesn't make pacts with weird extradimensional beings, at which point, things get a little funkier.
example:
The pcs are sitting at the tavern and placed an ad on the job board looking for help against a dragon.
DM: I burly unkept man with a red beard that would make a dwarf envious approaches your table. He carries across his shoulders quite possibly the largest Battleaxe you have seen. "I understand you are looking for someone to help with the Dragon."
Player: Yes, but first what class are you?
Most of our group consists of rpers but this one guy doesn't get the concept. Yes he is a heavy MMORPG player and not great at the RPing side of D&D. (Don't misconstrue this as all MMORPG players, but this one fits the stereotype MMO trying to convert to Pen and Paper.)
Like most people it really depends on the class, All the arcane, divine and most or the primal are fine with referring to class as a career. Really though it only bothers me when I introduce X NPC or DMPC to the group and I have one player in game ask what class are you expecting it to be above that person's head in neon letters.
example:
The pcs are sitting at the tavern and placed an ad on the job board looking for help against a dragon.
DM: I burly unkept man with a red beard that would make a dwarf envious approaches your table. He carries across his shoulders quite possibly the largest Battleaxe you have seen. "I understand you are looking for someone to help with the Dragon."
Player: Yes, but first what class are you?
Most of our group consists of rpers but this one guy doesn't get the concept. Yes he is a heavy MMORPG player and not great at the RPing side of D&D. (Don't misconstrue this as all MMORPG players, but this one fits the stereotype MMO trying to convert to Pen and Paper.)
I suppose the more appropriate IC response would be something on the order of asking 'what is it you do' or to modernize it, 'what's your skill set', since just wielding a battleaxe isn't necessarily an indication of class.
I personally enjoy introducing my Warlords as 'tacticians' or 'strategists'.
Dep In game...most classes work fine, fighter is a bit vague and people don't really go calling themselves rogues...
"Are you a robber?"
"I'm a thief!
For me it depends on the class.
Wizards,bard,sorcerers,monks,druids, cleric and maybe barbarians? Sure I can let it slide. I don't think I would do it, but I could let it go if someone else did.
If someone is asked to describe how their fighter appears in game to a party, because they are being introduced late or whatever reason, says "OK, well I'm a human fighter ....." it totally throws me out of game.
There are so many other words and better descriptors. To me a fighter makes it seem like you are a boxer, wrestler, or something. And even then there are better, gladiator for instance.
Fighter might actually be the worst for me, but warlord is pretty bad too.
As a player I prefer to be told that "Ok, my PC is a dark cloaked mage with long black hair pulled into a ponytail... whatever" or a "Feared lion warrior from the north, who carries a gigantic axe slung over his shoulder..." or maybe a "late middle aged scholarly looking dwarf, carrying tomes, potion vials, and a couple magic wands all slung into his various belts and bags." Sure if you give that description, you can go ahead and throw in "Also FYI I am a human swordmage, multiclassed into cleric", but please when asked to describe yourself in game tell me how you are a "Swordmage".