If the birth registration said Mary Elizabeth and baptism said Mary Ann, either the vicar / clerk made an error OR (a possible scenario)
Mary Ann / Elizabeth wasn't the person who took the baby to be baptised, but the adoptive parents. The mother may have given them the child not long after birth, and 18 months on they discovered there had been no baptism, but they only had sketchy information about the parents. They remembered her name was Mary something, didn't know the father's name but remembered being told he was a soldier.