Specifically, how mages refer to themselves. One person has fire magic, and has been referred to as a pyromancer, a pyromage, and a pyromagus. Is that overly confusing to the reader, to use the suffixes interchangeably, or should I pick one and stick with it? Are there examples of other writers who have used multiple terms, either to refer to the same thing or different 'brands' of magic?
Multiple terms meaning the same thing is fine but since they all start with the same root word, I think in this case it will be confusing.
I want to point out that, etymologically, a 'mancer' is a diviner: a practitioner of divination, not someone who can control stuff. People will definitely know what you mean if you call a fire mage a pyromancer -- that is the common usage -- but hey. I guess I'm just sharing factoids, really. It'd probably be fine. I might expect there to be some difference between these terms, but if the same person is referred to by all three, I'd get the gist. At the same time, I would think it'd probably just be easier to stick to one, honestly -- keep your terminology consistent. Do different groups or different individuals use different terms? Does everyone use all three interchangeably? How did the different terms come to be, in-universe? That's what I'd think about.
I believe thought that a pyromancer is a Mage that specifically deals and trains with fire or specialises in fire magic
If you want to be technically correct, @izzybot is right. It's an old term, derived from Greek. A pyromancer is defined as a person who tells the future by looking into fires, etc. However, it looks like some people are now using the word to mean a magician who controls fire. This is kind of like 'awesome' 'Awesome' used to mean something that inspires real awe in the beholder. ('Awe,' as in I'm so insignificant against the backdrop of the Grand Canyon, or in the face of this massive tornado.) Now it means 'good, tasty, clever, pretty' ...whatever. If it were me writing this, I'd maybe come up with another word, if you mean 'magician,' rather than 'diviner.' That way, the word pyromancer will continue to mean somebody who divines the future using fire. It would be nice to keep that meaning. There are no other words for that kind of divination, are there? Why not call your man a 'pyromaniac?' Oh, wait. That means something else as well... https://www.google.com/search?client=firefox-b&ei=YLHmWp3SOYmegAay-rmgCA&q=pyromancer+meaning&oq=pyromancer+meaning&gs_l=psy-ab.12..0l2.65464.66736.0.68530.8.6.0.2.2.0.70.386.6.6.0....0...1c.1.64.psy-ab..0.8.392...0i67k1j0i22i30k1.0.OO64vdEyOmY
What if they were different specialties within fire magic? Or perhaps different levels of mastery? "Yeah, Guido's just a pyromancer now, but he picks up fast, I can see his 'magus leveling him up to pyromage by the harvest"
I would be inclined to avoid the technically incorrect "mancer", because any of these names feel to me like the Latin name for plants--the more formal, academic, name. Someone outside the profession might refer to him as, for example, a fire charmer. If they're being academic and correct about it, it seems to me that they should be fully correct.
Another take on this could be cultural. Perhaps one culture refers to fire wielders as pyromancers, another calls them pyromages, and a another could refer to them as pyromagus. This could be further reinforced by having the same culture use a similar reference to another branch of magic. Like with water have hyrdomancer, hydromage, and hyrdomagus. Just keep people from the various cultures sticking to the reference they should be using for the sake of consistency.
I think you should pick one and stick to it unless you have a definite reason for having more than one term. Like, if one is a veiled insult, another is used for technical writing/formal occasions and a third is what the magic-users call each other, okay. But just random switches would seem sloppy, to me.