Amdapori White Magic and Mhachi Black Magic rely on drawing ambient 'external' mana from the environment (and from Void sources for the more advanced Mhachi Voidmages) to power their spells. The cumulative effects on the environment during the War of the Magi brought about the Sixth Umbral Calamity.
A number of White and Black Mages were among the survivors who settled in Gyr Abania to escape the flooding. Recognizing that it was their overuse of external mana that brought about the Calamity, the remaining mages worked together to develop extremely efficient evolutions of their magics that only utilized the caster's internal reserves of mana while minimizing the life-threatening effects of depleting their own internal mana through the use of a specialized magicked crystal as a spellcasting focus.
HEADCANON - The crystalized aether produced by the core spells (Jolt, Scatter, Scorch) of this new school of Arcane magic were found to have a distinctively red tint to them, due to the power source being the caster's own life energies - i.e. their blood. In an attempt to stifle unsavory labels such as "Blood Magic" or "Blood Mages", the practitioners of this new art would call it "Red Magic" and themselves "Red Mages". /HEADCANON
Thaumaturgy/Black Magic utilizes three of the elements (Fire, Ice, Lightning) in their combat magics, while Conjury/White Magic uses two elements (Stone, Wind) in combat magic and the last element (Water) as restorative magic for healing.Bonus question: according to you what is the element of Fleche and Contre Sixte spells? They say "delivers physical attack" but the only element red mage does not use yet is water aspected element.
If Red Magic utilizes the arts of both schools, we see that we use four elements for combat spells (Verthunder, Veraero, Verstone, Verfire) and one element for restorative magic (Vercure) - so that leaves Ice unused... until you realize the spectral blades of Fleche and Contre Sixte look and sound A LOT like the weapons Shiva materializes.