R.C. church also permits lay people to baptise in such cases. ( My classmates and I were instructed how to perform a simple baptism as part of an R.E. lesson at secondary school, in case there was ever a need. Some girls would have become nurses and midwives.)
At a time of high infant mortality doctors, midwives, nurses, family members, friends and neighbours would all probably be familiar with the ritual. A midwife, who, in a poor area may have been a neighbour, would have carried out many emergency baptisms of newborns not expected to survive.
The simple form of baptism which I was taught: No need for holy water. Words: " I baptise thee N.(if a name has been chosen) in the Name of The Father, The Son and The Holy Ghost." Also no need for godparents.
If the baby survived, parents might take it to church at a later date. There the priest would perform the rest of a usual baptism ceremony. Godparents would make promises on behalf of the child, there would be prayers, candle, white garment etc. The priest might carry out the pouring water over the baby's head and naming of the child, just in case it hadn't been done the 1st time. (I'm not quite sure about the last bit.)
After the ceremony, priest would enter details in baptism register. He would add a note explaining it was a supplementary ceremony. It might be in the form as for Elizabeth, above or other wording. A baptism register of 18thC I looked at last week had the words "supplying defect " (abbreviated) in such cases. Many babies in that rural parish were baptised by a doctor.
In Elizabeth's case it seems like her survival was uncertain from birth, so Margaret baptised her the following day.
I've read several enquiries like this on here in past few weeks.
Why don't you post the other Latin bit you can't read. Someone may be able to decipher and translate.
Have you seen the baptism entry of 8th May? What does that say? If there was a 2nd ceremony, there may be an additional entry between 8th May- 1st June. ( A baptismal register at an R. C. church in Burnley, Lancs. had some double entries. In those cases 1st baptisms were at workhouse.)