Including names and dates etc with your queries really helps...
Regarding births in institutions such as you describe,,,your information is close but not quite accurate.
What used to happen was single mothers who could not pay for medical care, hospital/birthing and/or to care for their babies and who chose to place them for adoption COULD stay with, receive medical care and have their babies delivered with (various) organisations/agencies..
Most mothers were never given the chance to see or name their babies as it was already agreed the babies would be placed for adoption,,,and USUALLY already had a family waiting to adopt.
The babies would simply be registered by the adopting family.
Sometimes the organisations did keep records of mothers names etc and date she gave birth and if child was male/female etc...
Which family got which child was usually recorded.
Baptism/christening records are available but not online in all areas for the years you are seeking. They would possibley be available in Archives and major libraries here in Australia, or may still be kept with church archives.
Problem is,,she probably wasn't baptised by her birth mother.
Findin the birth is what you need and it should be under the adopted families name...