Bio or birth mother implies a mother who has given birth but is not involved with their child or who had their child adopted.
This. If you've given birth to a baby but then given them up for adoption, or if you decided to act as a surrogate mum, your role WAS in their birth, but you no longer have any (or possibly very infrequent) part in their lives, hence you are accurately described as their 'birth mother'. It's a term that's clearly understood to mean that another woman has since taken over from you.
It's also a term sometimes used disdainfully and distancingly by (usually adult) children when they experienced abuse, neglect and/or extreme disinterest from their mother and they maybe have a step-mum (or aunt or older female friend) who has conversely shown them great love and compassion over years and whom they consider to have been more of a 'mum' to them than their actual mum ever was. This is their prerogative, but nobody else has the right to use this description on their behalf.
Either way, assuming that a mum is still active (or at least respectfully acknowledged) in her children's lives (or was until she died) and no acrimony has taken place, her relationship to them and significance in their lives has extended far, far longer than simply being responsible for their birth.
Calling somebody's actual mother their 'birth mother' makes as much sense as referring to your husband as your 'wedding husband'. Not strictly inaccurate, as he did become your husband at your wedding, but if you're still happily married, one would hope that the wedding was just the beginning of his marriage to you and not the sum total of its value. Similarly, you wouldn't refer to him as your 'first husband' unless you'd since divorced him and married somebody else, who had therefore now replaced him and taken over his old role in your life.
Theoretically, you could have any number of step-mums or step-dads throughout your life, but only ever one mum and dad, hence they get that simple title unless somebody else has clearly taken the role over from them or maybe they've behaved appallingly and proved themselves thoroughly unworthy of it.