I think I can explain what was happening here.
In Catholic belief and practice, baptism is a ‘once for all time’ sacrament. You only get it once and it is effective forever. No one can ever change that you are a Christian and a member of the Body of Christ once it is done. That’s why we don’t rebaptize people from other Christian denominations as, for example, the Baptists do.
Generally what happens when a child receives emergency baptism in a hospital, is that s/he receives the additional rites of the ceremony in a parish celebration later on–prayers, reading, participation of godparents, anointing with holy chrism, receiving of the baptismal candle, etc. But generally the actual dunking or pouring of water is omitted, since it’s already happened. If it is put in, then it would only be done conditionally–that is, on the condition that the first time it wasn’t done correctly.
I don’t know what happened at the parish ceremony for your son, but that’s probably why they said the second ceremony wasn’t real, which isn’t exactly true–it was a continuation and a completion of the first ceremony done as an emergency thing in the hospital.
Fr. Brett Hoover, CSP is the former Director and co-founder of BustedHalo.com. HE is presently a Doctoral Candidate in Theology at the University of California at Berkeley.