My kids are grown, and there are no step-parents in the picture (so this isn't a personal thing for me); but if there's one thing that I see online over and over again, it's step-mothers who refer to children's mothers as "the bio mother" or "the birth mother".
I know there all kinds of situations that involve young children or teens who have a step-mother. To me, pretty much the only time a child's mother should be referred to as "the bio mom" or "the birth mom" might be, for example, a situation when a baby's mother is dead and the step-mother has adopted the child.
Other than that, and (again, just my own opinion on this) no matter how much of a "loser" the child's own mother may be (or may only appear to be, for that matter; because it's not unusual for SOME mothers to have "a job done on them" when aggression and/or lies and/or ignorance get factored into an ugly divorce); and no matter how truly wonderful and caring the step-mother may be; as long as that mother (the real one) is alive she shouldn't be called "bio mother" or "birth mother" as if she gave up the child for adoption (unless, of course, she did).
Whenever I see some step-mother refer to a child/children's mother as "the bio mother"/"the bio mom" or "the birth mother", that suggests to me that the step-mother, for one reason or another, doesn't know, first-hand, the difference between "mother" (good, bad, or indifferent) and "step-mother".
To me, ideally, the step-parent relationship should be/could be a special one and a very nice one; but I'd more compare that "different but special" kind of relationship between step-parent and step-child as yet another "different but special" relationship children can have with someone like a grandmother or aunt. All of those "different and special" relationships can add to a child's life; but having known more than my share of kids who have been separated from their mother (sometimes for legitimate reasons, sometimes I've never been all that sure about); I just think it's easier for a child to under that his/her mother is, in fact, his/her mother - and he, as well as all the adults in his life understand and respect that.