I see where you're coming from. The situation you descibed would be how it would go in real life. We get matched. I call my mom and say "We're matched." "That's great!" "Yea, her name is Jane and she's really sweet... rest of conversation..." From them on I would just say "So I was talking to Jane and..." If I intriduced her to family or close friends I would say "This is Jane" and they would already know who she is. If I introduced her to a casual aquaintance I would say "This is Jane." and that would be the end of it.
As far as the "our" part. I have to say I don't understand where you're coming from in that respect. I can see why you wouldn't like "our bmom" because it's really not at all accurate. I think of our future child's birthmom as family, sort of like a sister in law (which works only if you like your sister in law I suppose, which I do). I'm very close to my sister in law and I consider her family, not just to my brother, but to myself as well. I don't have a direct family relationship with her(hence the in law), but my brother and niece do. I wouldn't refer to her as "my niece's mother" because, while that's important, she has an important relationship to me too. I'm not sure if that makes any sense. I just mean that our future child's birthmother will be my family too and I wish there was a word to describe that close family relationship to me too.
For the matched mom part, I was thinking more in terms of internet conversation where you don't always like to reveal someone's whole name. Also, when I post on the internet I don't want to have to say every time "the mom we are matched with" because it gets kind of wordy. That's why I was thinking of an abbreviation like our mmom instead of bmom or emom.