There's room for both, I think. As hinted at in the op and above, the more general/base the class, the more open for interpretation and different places in the world.
A thief is a rogue is a thief. Could be an acrobat guy with the traveling carnival. Could be an urchin pickpocket from the slums of the metropolis, or a thrill-seeking aristocrat's son cat-burglaring his way through the same city's upper crust. Could be a treasure-hunting explorer with a true academic interest for ancient artifacts or tomb-robber out for the cash-o-la. Could be a "scout" or "guide" between trading hubs, a celebrated locksmith and "vault builder" [trap-engineer], a "security expert" testing the defenses of well-to-do merchants' warehouses, and many many other bunches of characters - within the world - just being a "rogue/thief" on paper.
An Assassin, as a class, begins to limit those "in world" options a bit, as the "lore/fluff" of the class becomes more specific, as with it their mechanic niches narrowing as well. From the outside/baseline, they are mostly thiefly roguish stuff. BUt then there's some abilities and skills that a "thief/rogue" class doesn't have. So they are defined as an "Assassin" class...and that demands something of their lore. They know about poisons. Where/Why/How? They've learned some extra deadly ways of attacking people. Where/Why/How? There is a general presumption, if not stated outright in the class description, that you must have had some training -if not current affiliation- with some guild (or individual) who taught you these ways of "extra-sneakiness," disguise, and death-dealing. But there's still room for several character types.
Are you a current member of a guild or not. Reformed? Escaped? Did a rival guild/gang "off" your former crew? How many assassins guilds (or famed assassin individuals) exist in this world? On good terms with your former associates or being hunted before your betrayal of leaving? Are you an "apothecary" using your poison knowledge to form medcins and antidotes (and ya know, better poisons in your free time)? Are you a specialist for a government-sanctioned organization like Jame Bond? Military infiltration specialist? Are you just a flat-out demented killer? Freelance Bounty-hunter or "Problem Solver" for hire?
There are a LOT of places in the world you can exist. Not AS wide open as a basic thief/rogue, but pretty good array of options for creating a character.
Take it another step, and say there's a class called the "Shadoweaver." You're trained as thief, as get a trick or two of an assassin, AND you can conjure "shadow-stuff" weapons and move ("dimesion door") through shadows. You are the whispers of death on the wind. You are the fear of the chill wind in the night. Many believe you don't even exist and none, but the wealthiest of the land of Ick know how to find you.
NOW, this means you have to learn some magic, or be infused with a creature from the Shadow Plane, or drink shadow dragon blood from the Onyx Chalice of Ick, or... SOMETHING. In fact, within a single setting, it could be possible for there to be more than one way for your to become a Shadoweaver...but, if so, they are few, very specific things. There has to be more lore than a thief, or an assassin, to become a Shadowweaver. There is a more specific, specialized place within the world that the Shadoweavers hold...and it's not broad, and it's not for any/everyone.
So, basically, my take is, within a game system, and/or setting, there is room for any and all of these connections to the setting with a basic "equation," if you will.
From a baseline "general" class (this is the big four, I don't think anyone will dispute). The more features/abilities you add demands a commensurate addition to the "baked in" lore of the class. The more baked-in lore the class possesses, the more specific/narrow of a place the class has in the setting.
TL/DR: Could've probably saved us a lot of reading/time just using this example...
In my campaign setting world your going to find a whole lot of Clerics, to all different deities, different religions, different orders within the same religion, etc etc..
You are going to find significantly fewer "Paladins" than Clerics because there are fewer deities that empower paladins. Correspondingly, fewer orders if enough of them exist to make an "order" at all. Fewer, overall, places within the setting they can or would be found.
You are going to find significantly fewer "Whitehthorns of Gilea" (setting homebrew specialist cleric class) than Paladins because there is exactly ONE goddess who empowers one to become a Whitethorn in the first place. She is a goddess of Life/Healing, so those she selects are drawn from those worthy of particular blessing/boon from her order of pacifist healers for special training, different magical powers (than the typical Gilean cleric), and dispensation to engage in combat.
All three kinds of characters can be found and "have a place" in the campaign setting. But the more specialized/specific you get in class features, the more specific and limited your place exists within the world.