All three of my children are adopted in our family. Just not all by me. My DH step-parent adopted DS1. I refer to his birth father as bio father. Mainly because that is all he gave DS1. Biology. So, I guess in my mind that was always a slightly negative connotation...so I'm very interested in hearing first moms say they like or use that - although I can understand it - it's just all so personal. If I could, he'd just be sperm captain - since he never saw my son...that is the real truth. But, I felt that was a bit too callous.
As for DS2 and DD - both from Russia. We use Russian mom, Russian dad, Russian parents. I guess that wouldn't work the same for Alabama Mom or something like that...given the other options I would use first mom...but again, my kids were with those families for 5 years or so. So, they were the first family they knew.
The only term that makes me upset is real. We're all real...last I checked there weren't robots giving birth. It irks me when folks ask which of my kids is my real kid...yuck all around.