I always thought shaman was a generic term used for any priest-like figure in indigenous traditions.
Nope. It's a term from an actual religion that still exists, particularly in Siberia. For a while, anthropologists did kind of extend it out, in a "all primitive people are basically the same" mentality, but now it's reserved for religions that at least share the main characteristics of shamanism. Specifically, belief in a separate spirit world, which can only be accessed by specific individuals, who undergo ecstatic trances to get the spirits to intervene for their follower. There are only a few in North America, and I think six or seven in South America that practice shamanistic religions. Which, given the diversity of religions among the indigenous peoples, is to be expected.