I only use identifying terms (birth parent, foster parent, adoptive parent etc) when writing about these situations. Speaking to children in real life is a completely different issue - so I wanted to clarify it now.
General rule: When speaking to foster children about their parents, I generally use whatever they call them!I almost always refer to their biological/birth/first parents as "your mom" and "your dad". Occasionally, if they don't have a relationship with their parents, I will refer to them as "your Mommy -insert parent's first name-" or "Daddy Joe".
Very rarely do I use the terms "birth mom" or "biological parent" with a foster child - it just doesn't seem appropriate or necessary in most cases. I don't use the term "first parent" with foster children only because it seems uncomfortable with a child who honestly may have had 8 sets of parents - I don't want to get into numbering them all! Most kids know who I'm talking about by the context - if necessary, they'll ask for clarification.
I would never encourage or allow a new foster parent to demand that a child to call them "mom" or "dad". This is generally confusing for a child who already has a mom and dad. At times I have seen it also be traumatic for a child who is clearly attached to their biological parents and planning to return home. I generally call their foster parents, "your foster mom" and "your foster dad". The exception to this rule is if the child identifies their foster parent as "mom" or "dad". Then I will call them the same thing as well. It can be a tricky situation with foster children, but I usually try to avoid confusion by asking their foster parents what they are called before speaking with the children.
Most foster parents I've worked with go by their first name (Jane), their last name (Ms. Smith), or have a nickname that foster children use (Auntie, Nana, or even Judge - she really was a judge!). However, if there are other children in the home - foster kids will often slip into calling their foster parents "mom and dad" because its what everyone else is doing! I don't think this is a bad thing and shouldn't be discouraged. For all intents and purposes, that person is filling the role of "parent" for the child. And, for most children it is easier and calls less attention to their being an "outsider" in the family and to the outside world. I don't think it confuses children to call both their foster parents and their biological parents "mom and dad" if it is their choice.
Previous foster parents, that the children no longer live with, are called by their first or last names - "Rob and Sue" or "The Robinsons". When foster children are moving towards adoption in the home they're living in, they usually are already referring to their foster parents as "mom and dad" and so I do the same when talking about them. But, if we are talking about both sets of parents, I'll use the terms "adoptive parent" and "birth parent".
My bottom line is that, with foster children, I walk the fine line of not wanting to degrade their biological parents in any way and respecting the child's right to choose to call their foster parents whatever they feel comfortable with. I also encourage most foster parents to tell family members (and others that their foster children will come into frequent contact with) what their foster children call them - so that it doesn't create confusion or frustration during family get-togethers.
Domestic and International Adoption
I honestly don't know any children who were adopted as infants (domestic or international) that have contact with their birth parents and so I haven't run across this situation. But for the few adult adoptees that I do know, I use whatever terms they use - most call their adoptive parents "mom and dad" and their birth parents by their first names.
With the adults that I know - I use whatever terms they use. And if I don't know - I ask! I'd rather feel uncomfortable because I have to ask, than feel terrible because I used a term that was offensive to them or part of their family.
Hope that all makes sense - if you have a question, or want to point out a place that I could be more sensitive, PLEASE DO! I won't take offense! :)