I've explained the private re public many times - you are baptised just the once in church or elsewhere (for sickly infants not expected to live - and don't forget no modern medicine) at home - those that survived were then, some time later, received into the congregation - vicars/clerks noted either or both in the register. Sometimes, eg in Roman Catholic converts there would be a conditional rebaptism in case the 1st was deficient or in case of CofE converts from Quakers