How about I do you one better and show you that Summoners are Arcanists and Scholars are (for all intents and purposes according to Canon lore as you've decreed it) their own kind of mage?
From your own quote of Y'mhitra in that level 30 job quest (which i've verified for myself already):
Note that phrase "arcane practitioners". What does that refer to? Well, the most obvious option is Arcanists. They wield Arcanima, after all. "Arcane" is the root word Arcanima and Arcanist are both derived from too, which has linguistic significance here, and continues to be repeated elsewhere in the Summoner Questline (Such as by Principia when we first attain him during the level 60 Stormblood Quest, I'm combing further through for examples of this). There would be no reason to use the word "arcane" to describe them if they weren't supposed to be Arcanists. It's not any other kind of magic that they're using. We even have the Allagans linked to aetherochemical processes later on in the Summoner Quest line, implying that they're magical scientists, something that does fit under Arcanists in the lore, given they're primarily magical researchers and many are also tacticians. Hell even the Egi themselves are referred to as "Arcane" entities by Y'mhitra (specifically during the Level 58 Quest, where she also explains why we can't summon the other primal Egi and has us undergo "Trance" instead).We salvaged from the ruins a set of ancient texts describing the existence of a sect of Allagan mages known as “summoners.” These were arcane practitioners said to have the ability to siphon the essences of the primals and manifest this stolen energy as a biddable ally known as an “egi.”
Scholars, by contrast, are specifically referred to as Nymian Mages and are considered to be doctors off the battlefield and are given the role of field tacticians on it. Unlike that one 'arcane practitioner' line (where in the SMN job quests from that point onward they refer to those Allagan mages as Summoners, and only bring up the arcane when talking about the details of the actual magic they use), it's stated multiple times in the Scholar quests that they are mages of Nym. It's intentionally more broad. However, we know that they're definitively not Conjurers (This is explicitly called out in the level 65 quest), or Thaumaturges thanks to those classes and Scholar's respective historical roles in the War of the Magi, which in turn rules out White, Black, and Red Mages. What does that leave? Probably Arcanima, as there's clearly no connection from them to Astrologian either, but for all intents and purposes, we could treat them as their own kind of mage if SE wanted to.
Want proof? Watch this video of the initial parts of the scholar quest line (i'm avoiding his commentary here as it's not in the text). You'll note repeatedly there are references to Nymian Mages, not Arcanists. They do mention Arcanima, but only on the context of the player's skills. In the level 30 quest that Alka Zolka specifically calls upon an "exceptional Arcanist" so that we, the player, can lend our insight into magical research regarding Nym, something he lacks any understanding of as a marauder. Which again makes sense, as Arcanima is basically programming magic, and in the general scope of Arcanist's lore, they're treated broadly as researchers, tacticians, and scientists that all have expertise in various fields of magical theory, so Alka Zolka of course calls upon one for research and we can even assume that is likely what the Scholars were using. However, it's not explicitly stated here that Scholars did use Arcanima. Alka Zolka only brings it up in the context of noting our own expertise relative to his.
However it's also noted that the Fairy is a familiar bound to the Scholar Soulcrystal in order to preserve Nym's magical knowledge. In the level 35 quest Alka Zolka concludes that the reason you haven't learned anything (beyond Adloquium) from her is because she's forgotten almost all of it due to being inactive for roughly 1500 years since the era where the War of the Magi took place. The fact you've only recently attuned to one another (which happens basically just by you handling the soul crystal in the level 30 cutscene, as she's drawing out the remnant aether from the battle you had before to appear temporarily) means she can't pull out those memories out in order to help you learn anything. Alka Zolka assigns the player tasks to begin conducting experiments in an attempt to help her remember that knowledge, and thus pass it on to you, while he tries to decipher his texts to find anything. The result? He often finds general knowledge of their history, but he doesn't uncover any new spells. The reason we're using Arcanist abilities at all on Scholar at that point is because the Fairy literally can't teach us anything else. It's a case of them being the best fit for the scenario, and SE intentionally left the option to split them off later if they so chose to.
But yes, I don't have a basis for my carbuncle theory, beyond that literally Egi and Fairies occupy the same space as them mechanically and in terms of their respective job's lore they serve a similar purpose as a Carbuncle does to an Arcanist which they are given a direct link to, so I'll drop that. It seems obvious to me that the same base mechanic of creating an independent artificial entity for all three would be required or at least useful as a bridge to learning to evoke their respective pets but that isn't confirmed in lore. And they're all alike in that once brought out they have independent wills.
The only one that's directly shown to be different however is the Fairy, in that she's bound to the soulcrystal, something we definitively can say is not true of the carbuncles or the Egi. Fairies are stated to be familiars, who notably have long-term memories, which presumably is helped by the soulcrystal containing and preserving those memories. We do know that she at least requires a portion of our aether in order to summon her (this is specifically noted in the level 68 scholar quest), or at least a source to draw from, which is true of the Carbuncles and Egi. It is implied but not stated that in the first Scholar cutscene she appears in that she may have been using the remnant aether of that battle to show herself, but regardless of whether that was the source or not, she was only able to do so temporarily until we accepted her soul crystal from Alka Zolka afterword. She does have her own will too, otherwise that scene and many others don't work. We don't control her like we do the Egi. She serves us, but can act against our wishes without breaking our bond with her, which happens a few times in the Scholar questline.
While the Summoner Soulcrystal similarly contains memories of their art and thus has its use in their processes, you definitively require primal essence in order actually to summon an Egi using their methods. I assume not having a soulcrystal would similarly end in disaster for them because I think that was part of the plot point for that rogue ARR Summoner we had to put down to begin with, but we know for a fact you cannot summon any sort of Egi without first defeating a primal, then you must defeat the Egi you create from their essence in the appropriate conditions in order to 'tame' them. It's akin to domesticating a wild animal, which does also appear to be true of the carbuncles, though they're more akin to pets (otherwise the running joke about Tataru being unable to control hers doesn't work).
For Demi-Bahamut we know we needed to master Dreadwyrm Trance and Akh Morn in order to summon him, as that represented us learning to control and use his aether before we could even attempt summon him properly. I believe this had something to do with him being a far stronger primal than the usual Beast tribes but I don't recall the exact details, as Sari and Prin basically handed him to us as a reward for passing their tests. Apart from that, the minimum requirements appear to be the same as Egi (Know how to Summon, defeated the relevant primal), and it seems we have full control of him from the get go as well, and that along with being temporary we can say are likely features of Demi-summoning in particular.
Once thing I did note while looking at the scholar quest more closely (to ensure there's no more mentions of Arcanima beyond that video I linked), is that while the Fairy's memories bring out the core Scholar spells and Sacred Soil, that only lasts up until 50, from which the tonberries take over as your teachers from there onward. Which I forgot, admittedly. Notably though Surito Carito does explicitly teach you Lustrate, Indomitability and Dissipation in the level 50, 52, and 60 job quests you attain them respectively, both of the core healing Aetherflow abilities of Scholar and one of the tools that interacts with it. Everything else you learn in that period isn't explicitly from anyone. This doesn't override the idea that the Fairy and the Scholar soulcrystal are the ultimate source of the Scholarly arts, but it does lend credence to the idea that those abilities in particular are likely arcanima if we go by the same logic I was using for Carbuncles earlier, though again it isn't stated that they or any other abilities are when I looked at those quests. Excog is learned via level up and isn't referenced in Stormblood's questline as the quests are intentionally staggered differently to your ability progression there so that they're no longer locked to one another, aside from the capstone quest, where Lilac (the fairy in the Soulcrystal we wind up finding in the stormblood quests) is the one who teaches us Fey Union/Aetherpact. This is only stated during the text for unlocking Fey Union however, and to me that's even less 'Canon' than the Egi Glamours would be.
I do still think that Shadowbringers will confirm it one way or the other whether there's ever going to be a split off between both jobs however, with the default assumption being both stay tied to Arcanist, of course. Any lore they add to Scholar and Summoner on the First has the benefit of being from a separate world with different history, so it would be strange if they didn't confirm or even explore how Arcanima developed differently along the way in their respective job quests. I'd like it if they did more direct call outs to Arcanima too, to put things to rest, as it were.