I always get sci-fi/fantasy vibes from plane, but still like it. If you're targeting fantasy/sci-fi readers, that might be your best bet (since they're pretty guaranteed to know what it means). Same with "realm": very sci-fi.
I think, for the "average" reader who might not recognize the concept you're explaining, "dimensions" might hit the closest to home for them – they'll immediately know what dimensions are, and you can just flesh out that base knowledge and hammer in the extra facts you need: that there are three "main" dimensions, and that people can travel between them, and they each have their own properties/differences.
That said, honestly you go with any word you wanted (including an entirely-made-up one!) since it's probably such a base concept. Using a pre-existing word makes it easy for a reader to jump in and hit the ground running with understanding, but if you're unsure about the word (or want to use your own), you can always build that foundation with a little backstory as well. If you've got other parts of the plot you want to tie in to with a particular word (like layer), this might be the best way to go. :)