A good microcosm of this question is: how many different kinds of magic are there?
OO! OO! <raising hand> SIX! Arcane. Divine. Nature. Psychic. Eldritch/Forbidden, and "High Magic"/ Translevel.
When a wizard and a cleric both cast detect magic, are they doing fundamentally the same thing (the same essential gestures and words) with slight stylistic differences, to manipulate the same magical energies into roughly the same weaves, or are they doing two very different things: using two very different sets of materials to build different tools that happen to serve the same purpose?
The latter.
Mage [Wizard]: incantation, gesture, eyes glow. You see the auras of the magical things before you.
Cleric: invocation, mini-rite, eyes glow. Your deity has deemed fit to empower your sight with the ability of arcane magic. So, you see the auras of the magical things before you.
Druid: mini-rite, attunement to the pulse and flow of the natural energies of the world, eyes glow. You see the auras of things that do not adhere to the natural pulse and flow of energies of the world.
Psychic: concentration, concentration, eyes glow. You have attuned you mind to a frequency that is beyond the normal spectrum of human vision. You see the auras of items that fluctuate to a frequency that you know to be consistent with arcane (or divine or natural or psychic or unknown [eldritch]) magical energies.
"Warlock" [or whatever it becomes in later editions]: Your patron gave you this incantation and told you how to gesture. You do it. Your eyes glow. Your patron has shown you how to empower your sight with the ability of arcane magic. So, you see the auras of the magical things before you.
If the former, it doesn't make as much sense to split casters into discrete classes by where they learned. If the later, you need at least one class per type of magic present in the setting.
I would say martial classes should be about as broad as caster classes, which could mean either "just two" or half a dozen depending.
Mage/Wizard is to Swordmage/Spellsword (whatever Fighter-Magic-user split title you prefer) is to Fighter.
As Cleric is to Paladin is to Knight/Cavalier (non-magical but faith/ideal based).
As Druid is to Warden is to Ranger.
As Psychic is to Psi-warrior/"Jedi" is to Monk/Martial Adept.
As "Sorcerer" (something weird/unknown results in you having magic power) is to "Warlock" (something weird/but known is giving you magic power) is to Barbarian (something weird/unknown or known results in you having martial prowess other non-magical folks don't have).