Some more general terms I see as alternatives to "shaman" in the anthropological/comparative religion sense are "spirit worker" or "ecstatic practitioner", but these are certainly not as evocative for game purposes.
I mean, for that matter you could argue that "priests" are a mistake as a character class, because they really don't have a basis in Swords & Sorcery fiction.
From the point of view of being general and descriptive, "Fighting-Man" (as the term was used by military scholars and historians, and by pulp writers like Edgar Rice Burroughs) and "Magic-User" are very practical terms.
But they aren't very evocative.
The advantage of using specific historical terms like Shaman or Warlock or Druid is that they connote things. They carry or imply more meaning. They feel more grounded in some sort of place and setting, even if the definitions being used in the game are not well-matched to the historical realities of the terms.