I feel the need to come clean about my adoption language, although it's not like I've been hiding anything.
We use both "birth parent" and "first parent" in our house. Most of the time we just call K and R by their first names, so the labels don't really come up much. We aren't in many situations in which people don't already know who they are, but in conversations or introductions I sometimes use "birth parents," sometimes "other parents."
We have the luxury of being able to talk over language with K and R, so I use "birth parent" and their first names knowing it's what they prefer. I don't know how Puppy will refer to K and R when he's older; that is up to him. Nor do I know how we will refer to our second child's parents, since we haven't met them yet. Although K uses "birth mom" for herself, she refers to her first mom as her "biological mother." That fascinates me.
I talk about the people who might look at our profile as "parents considering adoption" or "expectant parents." Our introduction letter just starts out, "Hello."
I think "real parent" is pointless as a label, since it describes both sets of parents.
When I initially heard "first mom," I didn't like it much. It made me think of "first wife" and "second wife." That made me think of divorce, in which one person stops being the wife and the other starts, and I didn't like the connotation of K's motherhood stopping. It also made me think of polygamy, where there is generally a hierarchy among the wives and I didn't like that image either (especially since I was #2!).
I use "first parent" when I write not to make a statement (although I don't mind if it does). I started using it out of respect to those who are troubled by "birth parent." Over time it has grown on me and I actually prefer it now. K and R were taking care of Puppy before T and I ever knew he existed. They created him, they determined who his everyday family would be. They continue to be his link to his origins. "First parent" encapsulates those things for me. I'm identified as "mom" every day by friends, strangers, family, and Puppy himself. What is the harm in acknowledging that I'm not the first one to hold that title?