It could be every show had its own conventional name for the concept.
Nope, in my experience the terms tend to be used pretty interchangeably by any given series -- various mixes of "parallel/alternate" with "world/universe/dimension/reality." It's not like shows would have style guides for something like that. It'd just be whatever the screenwriter chooses to call it.
By the way, via a phrase search of Chakoteya's transcript site: TOS used "parallel universe" in "The Alternative Factor" first, "Mirror, Mirror" second. TAS use "alternate universe" in "The Time Trap," and DS9, VGR, and ENT used both phrases interchangeably. VGR used "parallel reality" once; DS9 used "alternate timeline" once. ST 2009 used "alternate reality." "Alternate dimension" showed up in VGR twice, but I don't think it was used to mean what we're talking about here. Doctor Who
has used both "parallel world" and "parallel universe" multiple times. Surprisingly there are no hits for "alternate history."