That's kinda debatable.
There's no "traditional" blue mage, they're all fairly different.
In FFV, blue mages can use knives, swords and rods, they're restricted to light armor and have a penalty to str and a high bonus to magic, they are casters first and foremost.
In VI, Strago uses rods and maces and wears robes, he is a traditional caster.
In VIII, Quistis uses Whips and Blue Magic is her Limit Break only, first one that's not a "traditional caster", but also isn't a traditional blue mage since blue magic is limited in use.
In IX, Quin uses Forks as weapons and can wear light armor, Blue magic is learned by eating enemies, can't say much about this one since I haven't played it.
In X, Kimahri can uses Lances and Armlets as armor, and uses Blue Magic during Overdrive(limit break system iirc) only, definitely not a caster, however Blue Magic here feels a bit secondary due to being tied to LB.
In XI, Blue mages can use swords and clubs(swords being the better option but clubs being generally the "caster" option if you make a pure caster build), they can use a variety of armor including heavier armor and they can be played in a variety of ways, including tank, sub healer(maybe main healer on easy content), melee damage dealer or magical damage dealer, so caster and not caster, depends on your choices, mostly played as a DPS or tank in current content but does caster for soloing some stuff for example.
In FFTA, Blue mages use Sabers but are restricted to light armor, as far as I remember, they're mostly casters and don't use their sabers for actual combat.
Overall, there's a fair amount of variety, but there's plenty of "traditional" Blue Mages being purely casters, both thematically and in practice. Because of the nature of the abilities granted by Blue Magic, a bunch might seem less "caster-y" than other stuff, but they'd still count as casters.
While Blue Mages are always somewhat different from each other, we can see rather clear lines of commonality between many of them - with Strago being the only major outlier here.
Of all of the examples you listed - which is pretty comprehensive - almost all of these directly put Blue Mage in the hybrid category. One very important aspect here, which you'll see come up a good few times across the series with this job especially, is that it's designed in a way that allows it to be complementary to a variety of different types of character archetypes. Blue Magic is, fundamentally, something that behaves in a strange way compared to the straight-forward nature of other types of character ability sets, and this aspect allows it to complement a variety of things in different ways.
In FFV, the selection of weaponry available makes a bigger overall impact to their combat performance than their innate stat bonuses do. This allows them to function quite well as a hybrid physical/melee character, as it's quite easy (and intentional, for sure) that they're able to strike a middle-ground without being useless on either front - despite what the strength penalty might imply when looked at alone.
Strago is easily the only case where Blue Mage has appeared as strictly a pure caster, as he is simply not well-equipped to do much else.
Quistis, Quina and Kimahri are intentionally designed to be somewhere between hybrid and physical - and notably, characters in Final Fantasy VIII don't have a high degree of variance in their stats and equipment to begin with (outside of some outliers, like Rinoa), as the game's primary means of character differentiation come in the forms of limit breaks and junctions. Kimahri and Quistis have their access to blue magic strictly tied to limit break mechanics, whereas Quina has weapons access that is on par with most of the game's physical attackers. Notably, Quina's weapons have a high degree of randomness to their damage variance, despite that the base attack power is still twice that of caster weapons like what Garnet, Eiko and Vivi use. It deserves mentioning that Kimahri is generally considered to be bad at everything due to the game's scaling, but the intent is that he's able to be flexible and easily go into many other characters' specs due to his place on the sphere grid, giving him an ease of malleability with which he can go toward physical or caster at the player's whim - this is kind of an important part of the concept, even if they did miss the mark balance-wise pretty egregiously.
One you missed, Gun Mage, is the FFX-2 equivalent of Blue Mage. It has enemy skills available to it, as well as the ability to scan enemies - and statistically, it's something of a powerhouse in having high both strength and magic stats, but it low on defense.
Blue Mage in FFXI is exceptionally-well-designed and highly-malleable in the way. As mentioned before, it's that ability to adapt to a multitude of functions and playstyles that causes this to really shine through as probably the most definitive interpretation of what a Blue Mage really is.
In FFTA and FFTA2, Blue Mages are back to being hybrids again - stat growth for Blue Mage is almost strictly even across the board, outside of being a bit on the slow side. Notably, most of the Blue Magic available in both of these games are support-oriented, with many of its spells being things you can't use in such a direct manner as more typical kinds of magic. Again, this lends it exceedingly well to function in a supportive way to many different kinds of other jobs as these are games with job systems in them. Damage > MP is additionally one of the most potent reaction skills in the game... specifically for physical combatants.
So in the end, the common theme tying Blue Mages together isn't simply the fact that they can cast enemy skills as magic - but the fact that (outside of Strago, who is again the outlier here) the unorthodox nature of spells they learn, combined with the equipment they can use, allows them to functionally be very versatile and complement a variety of different character archetypes in different ways. I fully understand that would be difficult to implement well in something so rigidly-structured as Final Fantasy XIV, but let's not pretend they couldn't have embraced it in a way that could've been a support-DPS-melee-caster, which would still solidify it as having a justifiably-unique place amongst the game's array of jobs, while also satisfying fans of the job itself.
