I'm trying to remember where I read a short story that I read around (I think) 1991, though I think it the collection was older than that. The idea was that each word ending in -verse was a sort of cosmos. The narrator pitied us, as our "universe" was so named because each consciousness was separate, whereas for example in the "diverse" there were exactly two individuals - I think the phrase was "allness was twoness" - but I don't get anything relevant when I google for that one.
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2There's a story in 'The Knights of the limits' by Barrington J Bayley in which a chessboard is taken over by travellers from a universe of discreet (non continuous) space who describe other universes based on different numbers... I don't remember them naming them differently, so this might not be it, but worth a look. Unfortunately I can't find a detailed description online, and I'm not sure which of the stories in the collection it was - a very long time since I read it, and I'm not at home to check my copy at the moment– Happy RobotCommented Apr 30, 2018 at 14:57
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I've had a chance to look at my copy of the Knights of the limits, and despite some differences I'm going to propose this as a possible answer: 'The Exploration of Space' by Barrington Bayley (the first story in his collection 'The Knights of the Limits'
During an opium dream the narrator is visited by explorers from a universe where space is non-continuous who manifest by manipulating the pieces on his chessboard (because pieces on a chessboard exist in discreet positions and cannot exist part way between squares, the chessboard provides a space in our universe that they can interact with) Most of the story is a discussion of different kinds of universe based on different physical laws and speculation about differing kinds of consciousness in those universes.
It doesn't contain the exact phrases mentioned in the question and does not give the universes different '-verse' names but it does have some phrases that could be misremembered over time:
"Eagerly I besought him to tell me as much as he could of this vast universe of divers space-time..."
When talking about a universe based on the number two: "Two-ness is ideal, and singleness is incomplete in the same way that a fraction or a part is incomplete in our world"
Sorry I can't find any online description of the story,
answered Apr 30, 2018 at 19:06
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That does sound surprisingly close, but the use of the "verse" suffix is the defining characteristic of the story I remember.– SimonCommented May 1, 2018 at 12:45