This is something I've been kind of wondering about too, what words most appropriately convey the correct meaning. (As a writer, I tend to obsess over words.)
When I was in a relationship with J (female) previously, I called her my girlfriend, she called me her girlfriend, and it sounded right.
Now that I'm in a relationship with Morp (male), sometimes I call him my boyfriend. When I came out to one of my clients (and told my parents about him) that's the word I used - it is the most popularly accepted for a male romantic partner - even though it seems sort of trite. I do refer to Guy as my husband in vanilla settings.
In fetish/bdsm circles I tend to refer to Guy as my Master, and Morp as my Dom (even though Morp considers himself more of a switch or kinkster and being called a dom is not exactly his cup of tea; he's dominant with me so it's as close to accurate as I can get in one word). Since he doesn't prefer the term dom, he requested I list him as "lover" on our fetlife profiles. I agree with the concept that lover should really be the most accurate, as it refers to LOVEr, but unfortunately, it has the connotation of a primarily sexual partner. Which sort of annoys me, because I think the love part is more important than the sex part.
When I refer to them in plural I just say "my men", which works in plural, not so much in singular, though I could see it working with females - "these are my ladies" or "my lady". "My man" has almost a misogynistic ring to it, to me; so while I appreciate the sound of it (I think it sounds possessive) in private, it sounds funny to me in public.
In places where I'm not out I usually use "dear friend" to specify him as a friend who is closer and more personal than friends in general.
I really like the term beau, but I dislike how it is pronounced, and the fact that most people don't know what it means offhand. If I'm just writing as opposed to speaking I might use it. Similarly, SO is neat shorthand, but clunky to say "significant other" and it sounds somewhat distancing.
Finally, I have always liked the use of "fiance/e" to indicate one whom one would get married to if it were legal (ie when same sex couples use it, or if for example Morp and I ended up in that level of commitment, since I'm already legally married to Guy) in addition to for one whom one will get married to; as it quickly conveys the level of 'seriousness/intimacy/entanglement' of the relationship.
Guy - Dom/husband
Minx - sort of dating
Cru - (no longer dating)
Lee - Guy's girlfriend