I've been trying to work my way through a thought about the three current Healer Jobs in FFXIV, mainly because I keep seeing questions about their differences from interested players in various discussion forums. Healers are not the only ones, with Tanks also getting a lot of "which one is fun to play" questions, but I'm less familiar with those.
I suspect a lot of issues with pinning it down comes from a mid-tier fact about FFXIV's gameplay. Specifically, the base-tier fact about the Healer role is that they prevent defeat through keeping HP up. The mid-tier fact is that doing more damage is more valuable in the long run, so Healers actually want to heal as little as possible while still keeping everyone alive, ie at minimum of 1 HP at all times. Taken to extremes, any HP above that single HP is a buffer zone for whatever the next fight mechanic is, apart from certain specific mechanics that require full HP. Obviously, depending on player skill, the range of that buffer varies, so it's still a good idea to tell newer Healers to keep people topped up. But by strict optimization, every GCD spent healing is a GCD spent not contributing to damage.
We all know how to characterize White Mage: it's the Power Healer. Almost everything is bent towards the task of restoring HP, through sheer healing power. It's the baseline gameplay for the Healer role, although the efficiency of Regens compared to straight heals means there are ways to optimize even this straightforward playstyle. However, a lot of its HP restoration is based on GCD heals, hence Regens being considered more efficient. Its attacks are also mostly GCD (with Assize being the sole exception), and like its heals, potencies are pumped way up. Basic WHM gameplay is essentially brute-forcing the role of Healer through sheer power.
I'm not as familiar with Astrologian, since I've not had much experience with it in gameplay above 60. (All my AST levels above that were from Alliance Roulette, before Rabanastre was added.) It seems like it's the buff healer, as in a Healer that contributes via buffs. Other than the card mechanics, it can be a discount WHM or a discount shield healer (I'm not saying "discount SCH" for reasons I'll get into later), in that it doesn't quite get WHM's other personal support tools for its own potencies, even as it gets a few for itself.
Overall, I'd call AST the Strategic Healer. Apart from the straight heals and damage abilities, every other action you perform needs to be considered in service to whatever the situation demands. You can Lightspeed for more healing output at the cost of damage, you can put Synastry on either your primary healing target for a secondary one, and the cards are obvious; once you draw a card, you need to decide what to do with it, depending on what other cards you have drawn and what you did to them, rather than blindly follow a flowchart of This Card Means This Action.
Which leaves Scholar, and the main impetus for this post.
SCH gets labelled as the "shield healer", which I think it fairly inaccurate. The primary shield spell, Adloquium, isn't actually very powerful; without a crit, it generally lasts a couple of GCDs at most before the shield falls off. You can spread it to the entire party on a long cooldown, and if you get a lucky critical Adloquium you can ignore several mechanics, but that requires excellent luck and timing. Nocturnal Astrologian gets a slightly more powerful single-target shield, again not assuming Critlos, since it's not like you can plan for those outside of fishing for them.
However, from levels 30 to 49, the only real difference SCH has from WHM are the shields, and the fairy. So from the point of view of someone who just started out trying SCH, SCH is indeed the shield healer, or possibly the pet healer. The fairy just heals automatically (and relatively invisibly), so people tend not to notice it in first impressions.
At level 50, though, everything changes for SCH. Suddenly Aetherflow makes sense for a Healer, rather than being a holdover from Arcanist going into Summoner. Lustrate, and later Indominability and Excogitation, illustrate a whole new identity for the Scholar that was far in the background during 30-49.
Personally I would call SCH the Off-Global Cooldown Healer. The bulk of its healing identity revolves around either using the fairy, or using the Aetherflow charges for oGCD heals. If someone is taking damage, just toss a Lustrate on them, and continue using GCDs for whatever you were doing before, which is probably going to be contributing damage.
The use of shielding in this case is therefore not so much to outright prevent damage, but to delay the need for other healing, including both GCD and oGCD heals. You don't spam Adloquium to keep a shield up; you use it a few times to let Excog last a bit longer, maybe a Fey Union if needed, and hope that the Tank's HP doesn't fall so fast that your three charges of Aetherflow per minute (or per 45 seconds at higher levels) cannot keep up.
This is also why SCH gets a reputation of being a Green DPS; despite WHM's higher damage potencies, SCH can afford to spend GCDs simply casting damage spells, rather than worrying about healing spells.
The fairy herself is also best thought of as a placeable single-target healing totem. She can take some of the burden of healing off our GCDs, thus allowing for even more "free" GCDs to do damage.
So Scholar has the issue of changing identities multiple times in its levelling process: Arcanist is a straight DPS that can (but really shouldn't) heal from 1-30 (Mind and Intelligence are equally present in low-level caster gear, so stats are not an issue; again, it's GCDs). Scholar is a fairy healer from 30-49, with shields as their powerful equivalent to Cure II and Benefic II. And at level 50 and above, Scholar is the oGCD Healer.
Which explains why new players wanting to be told how SCH plays have to be told that it's a "shield healer", since that's what they'll experience for twenty levels as their first impression. Which may cause some problems when they hit level 50, and still think that it's a "shield healer", and not use their oGCD abilities as much as they should.