Live birth refers to the birth of a living child, and the term is used to contrast with stillbirth. Explicitly, it means that the fetus, whatever its gestational age, exits the maternal body and subsequently shows any sign of life, such as voluntary movement, heartbeat, or pulsation of the umbilical cord, for however brief a time and regardless of whether the umbilical cord or placenta are intact.
We say that the live birth is to a particular woman if the maternal body being referred to is her body. The woman is considered the "mother" as far as the recording of the live birth goes. She need not be the genetic mother of the baby. She also need not be the person who takes care of the baby after it is born.