Old post ,but still accurate.
https://boards.na.leagueoflegends.com/en/c/developer-corner/3A5uuBw7-champion-subclass-list
The link he uses is broken due to age but it was copied and pasted on a Wiki of Riots official stance.
They make it clear that when it comes to categories, not everyone does or will agree, but it's what they settled on.
As far as "Mage" goes here is the following.
Mages are champions who typically possess great reach, ability-based area of effect damage and crowd control, and who use all of these strengths in tandem with each other to trap and destroy enemies from a distance. Specializing in magic damage, often burst damage, and therefore investing heavily in items that allow them to cast stronger and faster spells, mages excel at chaining their abilities together in powerful combos in order to win fights, though their abilities also tend to be difficult to land and can be mitigated, if not avoided completely, by their targets if they react in time.
Though mages tend to focus on killing prime targets in combat, they can also fall back to their innate crowd control and utility to manipulate key opponents, protecting their team from them or setting them up for a takedown, and in the right circumstances can damage and control multiple enemies at a time. In spite of the influence they exert, mages tend to be innately fragile, and fall quickly to direct strikes.
In general, mages are capable of dealing well with marksmen, as their burst can kill them before they can return the same amount of damage, and fighters, as their crowd control tends to make them excellent kiters. However, they are easily shut down by assassins who can often bypass their reach and spells completely, and tanks, who can lock them down and soak up their abilities better than other classes. Sub-classes of Mages are burst, battle, and artillery Mages.
The sub class for Soraka and Janna and those others you spoke of that feel they fall under that category are "Controllers" class not "Mage" and sub class Enchanter or Catcher.