I’ve never heard either ‘partner’ or ‘spouse’ used to signal inclusivity, and I don’t see how they would do that. LGBT people have boyfriends, girlfriends, husbands and wives, just like straight cis people do. How would any of those words be non-inclusive?
Around here (Ireland), ‘partner’ means ‘It would feel silly using boyfriend/girlfriend at my age and/or for a relationship of this seriousness’, and I’ve never heard anyone use ‘spouse’.
In Australia, “partner” suggests a committed conjugal partner; partners may or may not be married. (It also suggests a business partner, but it’s usually from the context which sense is invoked.) “My partner”, in the romantic/conjugal sense, could be of either gender, and we may or may not be married.
A partner to whom you are definitely not married is your “de facto”. Again, this is gender-meutral; if I mention “my partner” you don’t know whether that’s a man or a woman, unless you already know my orientation.
A partner to whom definitely are married is either your husband or your wife, depending on the partner’s gender (and regardless of yours). “Spouse”, I think, is only used in the plural, in contexts which suggest both husbands and wives.
“Signficant other” isn’t used, in my experience, except humorously. The two-adjectives-in-search-of-a-noun structure just screams mealy-mouthed euphemism.