Interesting responses. I think I like the explanations about how a specific type of magic is used mattering most, but I disagree still.
I mean; if you use a smithing hammer to hit someone in the head, does it turn into a warhammer suddenly? It's still the same item, though it's way of usage changed, the hammer is still the hammer.
Priests started to use the light first. They then taught it to knights/warriors/fighters which combined it with armed combat => paladins. Hence, calling paladins battlepriests sounds not wrong to me.
Warlocks used fel first. Demonhunters infused that warlock magic into their bodies to strenghten themselves, but it is still warlock magics nontheless. Illidan consumed Guldans' Skull (a warlocks' essence), obtaining warlock magic to enhance his martial abilities.
Mages used arcane first. If warriors started complimenting their fighting with it, we would get battlemages, or spellbreakers, or whatever. They would still be mages though.
Shamans used elemental magic/shamanism first. Monks, Druids and Death Knights seem to use either spirit or decay as basis for their magic. They developed special niche skills within the wide spectrum of shamanism, but it's still shamanism at the core (if my theory is correct, anyway).
Shamanism also already includes many different classes already; Spiritwalkers, Geomancers, Farseers, Oracles, Witchdoctors, Shadowhunters ...
It is really an umbrella term for spirit magic. Since I would confuse people speaking of spiritualists though, I decided to stick to Shamans.
As for warriors/hunters: You are thinking in in-game terms, not lore ones.
Lore-wise, the tauren civilisation for example has included the great hunt into a ritual of initiation for every coming of age Tauren. Even if they decide to choose the axe, the totem or the pelt later on, they are still trained hunters, lore-wise.
Like-wise, any tauren trains self defense growing up, and can be considered a reserve warrior, even if they decide to become merchants later.
Ret/Prot paladins, Havoc/Vegencance Demon Hunters, Enhancement Shamans, Monks do basically the same a warrior does, but enhanced with magics. If you go by lore, a regular warrior is inferior to a magic using one. Again, not related to ingame class balance and such.
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And blizzard is ice magic => water magic => elements
In other words, there is bound to be more game developing companies ment to complete the circle!
Sad that mages cant spec improved blizzard anymore to solve our problems.