I am 100% behind making classes feel different.
I am 150% behind making casting classes feel different from each other.
I am 173% that there is too much/many casters doing too much casting in 5e.
AND, I am 187% -maybe 200- having different casting classes, not just feel, but play differently. Not in narrow-to-meaningless ways, or necessarily mechanics (though those certainly matter/contribute), but have different expertises/areas they excel.
Different casting level caps by classes could, in fact, be a good part of that third "difference" making.
I think the way you are differentiating the classes works pretty well...and is similiar in some respects to my own homebrew breakdown. I think weeding out Bards and Warlocks as their own, most limited casting, but with more (or most?) explicit supernatural powers/"spell-like" abilities, AND the addition of a tertiary (would be quadrilateral, in my system/world) "Occult" power source is a very exciting innovation.
For me, first off, there are Mystic classes -whose primary feature/difference from other classes is their capacity to "Channel [Source/Energy]," whether that is Divine energies, Nature [Primal] energies, or some other source -which could easily be defined as Arcane (or "Occult"). Both Bards and -were I to include/have one- Warlocks would fall into the Mystic class. This also lets them have abilities that are not "magic" based/dependent, like some combat ability, some armors more weapons than wizardly mages would be allowed.
And, there are Wizard classes -whose primary feature/difference from other classes is their capacity for "[Arcane] Magic Use," as in spells, ritual, magic item use, and creation/expansion of all of those.
In my game, spell progression for Mystics are a "secondary" feature, not kicking in until 3rd level. So progression is, and remains, behind Wizard classes of the same level. The Mystics' primary magics are from/through their channeling. They are imbued with the power that permits them to do supernatural things/effects. Upon advancing levels, learning to use that power to "fuel" spells is a skill/byproduct of that power with which they are imbued.
I see a very clear (and easy) narrative/flavor difference that you could separate out the Bard and Warlock (and probably other similar archetypes) in that, whereas a Cleric a Druid is a "conduit" for their power source: e.g. the power comes into the caster and flows out through the channel or spell effect; they in the true sense "invoke" their power, and the power does what the cleric wants.
For an "Occult" class, the Bard/Warlock is more of a "battery." They "channel" in a way, but that is more that the power comes in to them, and kinda just bubbles/boils there for a time, until the Bard or Warlock "spends/expels" it for some effect they can do, whether that is a magical bardic song-channel, a warlock's impromptu "patron power/feature," or casting spells. These classes do "invoke" their power (either from the "Cosmos/Divine Music/Song of Creation," however you want to flavor the Bard's source, or a Warlock's [arcane] Patron -assuming that base flavor structure for the class remains), but they also, then, specifically "Evoke" that power which they have/store. As they advance level, they -like all casters- have more power to use for different things...and some of those things can be spells and higher levels of spells, but it needn't be the only supernatural way outpour their energy.
The Wizards then (Mages, Illusionists, Psychics, Witches, et al, for my system/setting) are specifically, and accurately (rather than a narrow "specialist school"), "evokers." They use incantation, ritual, or device to pull the energies (magic), directly, from the universe/cosmos in specific ways to elicit specific (reliably repeatable) effects. (Whether you want to fluff that cosmic energy as stemming from a "Weave" or some cosmic "radiation" inherent to the world's material universe or born of unicorn sneezes and pixie farts, what have you, etc...).
"Wizardly" casters figure out (through learning or innate ability, and really, in almost all cases a combination of the two - making the wizard/sorcerer split less than unnecessary, imo) how to shape the raw energy to do things they want.
The "Mystical" invoker-casters are granted that energy "pre-shaped" as it were.
The "invoker/evoker" combo "Occult" casters are granted [or siphon off?] the raw energy (whereas the full evoking "wizards" have to grab/grapple/harness/direct the energy themselves, Bards/Warlocks are "given/granted/gifted/imbued" with it), store it/build it up, and then figure out how they can/want to shape it.
They are the magical "cheats," as it were. The guys copying their neighbors' work. Finding the "shortcuts" to power instead of the intense training/study/effort of the wizards or devotion/enlightenment/spiritual fortitude of the mystics.
And, yes, I don't see why -if they are being "balanced" from their lack of higher level spells with additional supernatural abilities, channelled effects, and/or innate powers- then where's the harm in limiting their casting caps? I might even go further than the initial proposal...as spell power levels on a scale of "1-9" are really divvied up in triads: 1-3, 4-6, and 7-9. Adding in "cantrips/0 level" you could argue there's 10 spell levels.
So, if you're going to break the casting classes up, they should be broken on those lines, I'd say.
Mages (arcane), Specialists (specific arcane), Psychics (mental), witches (combo): Cantrips - 9.
Clerics (divine), Druids (nature), shamans (spirit world/combo): 1 - 6, begin spell accumulation at 3rd level.
Bards (nature), Warlocks (arcane), artificers(?): Cantrips - 5, cantrips at 1st level, begin spell accumulation at 3rd (or 5th?) level.
I don't necessarily use them or -in the Ranger's or Monk's cases allow them spells at all, but if you are using spells or granting supernatural effects, the flavor/power equality should be, roughly:
Paladins, Rangers, Monks (argument could be made for eldritch knight and arcane trickster types here, too): 1-3, begin spell accumulation at 5th (or even 7th?) level.
I'm really liking this direction.