To address the classic Final Fantasy WHM....
White Mage was Final Fantasy's answer to the more traditional "Priest" of Dungeons and Dragons (Something not associated with religion). All-in-all, the job is largely associated with very standard "holy magic" themes. Their most common spells include:
- Cure/Cura/Curaga (HP restoration)
- Esuna (Status removal)
- Raise/Arise (KO revival)
- Regen (HoT or HPT [heal per turn])
- Protect (Physical Defense)
- Shell (Magic Defense)
- Holy (Holy element damage)
- Dispel (Enemy buff removal)
Beyond that, other spells are more varied from one game to the next. You'd typically find some cases of stronger or weaker versions of these spells at different level thresholds, buffs and/or debuffs that sometimes gets shuffled around between being Black Magic or Time Magic (Haste if there is no time mage in the game, for example), and niche alternatives based on the game. They've never been a nature mage. Only in Final Fantasy III did White Mage have an element other than Holy, being Wind. Aero spells were classified as White Magic (Then later Blue Magic in FFV). This is because Final Fantasy doesn't really respect Earth, Wind, and Water in the way it does Fire, Ice, and Lightning. Sometimes Black Mages have them, often with completely irregular distribution, and other times they don't.
FFXIV's take on White Mage, as least thematically and aesthetically, is entirely unique. This seems to be largely based on the adaptation of 1.0s original complete departure from the traditional jobs that literally no one asked for and was met with heavy criticism. In order to salvage the lore they built, they needed to use Conjurer somehow, so they made it the precursor to White Mage and changed how Conjurer and Thaumaturge worked from 1.0 to be miniature White and Black Mages respectively.
Geomancer, on the other hand, has never been a healer, at least not when named. Traditionally, Geomancer's role was as a budget Black Mage basically. Geomancy costed no MP and would generate one of several spells selected at random depending on the location you were in. Fighting in a damp cavern filled with pools of water? Expect water and ice attacks. Interestingly enough, they always had a small chance to cast Shadowflare of all things in Final Fantasy III marking the first and only time this spell was used by the party for a very long time. FFXI was the first time it transitioned into a support role, though as a buffer/debuffer with offensive magic. Their only healing there was a regen and an MP refresh I believe, but I'm not a FFXI expert so I might be forgetting something.
That said, despite FFVII not having a job system, some of the characters were thematically associated with a job from a past final fantasy game:
Cloud - Warrior
Tifa - Monk
Yuffie - Ninja
Cait Sith - Gambler
Barret - Machinist/Engineer
Cid - Dragoon
I have no idea what Nanaki and Vincent were supposed to be based on, but Aerth was themed around a Geomancer--a healer that spoke with the planet. We actually see her staff in the White Mage's LB3 in FFXIV. You can really see a lot of the allusions in the names of her Limits: Healing Wind, Breath of the Earth, and Planet Protector.
All-in-all, I'm rather happy with White Mage's aesthetic. I get why they made the change to Glare and Dia in some ways, and why Holy was there from the start, but I'm happy that even if Stone and Aero are gone, that they are retaining a heavy floral motif. Flowers, in a way, can be seen as a representation of Earth, Wind, and Water. Flowers need good soil, water to grow, and wind to carry their seeds.
That all said, in regards to OP's comment about Geomancer, we will never be getting another split-job situation ever. There are a few things Yoshi-P has vehemently stated will never happen in FFXIV, and one of those is making another split job. Summoner and Scholar being joined at the hip has done nothing but give the design team a major headache. People hated Summoner being a DoT mage in ARR and have continued to complain about its lack of job fantasy even up till recently until finally this new Rework delivered on what Summoner fans have wanted since day 1. Meanwhile it made Scholar a monster of a healer in HW that had no equal, and they had to massively gut it over and over to try and bring it down, and look at it now in tatters.
Geomancer may one day be a job but there's no reason to staple it onto the Conjurer when they can just make it independent like everything else. It will be a long time before we get another healer though. It's possible though that they might make it a DPS. We could see ourselves introduced to a branch of Geomancers that have turned to war unlike the ones we're currently familiar with. It's hard to say for sure.