Since the child is not baptised at the second ceremony, there remains the question of how to mark/record the occasion. I haven't found directions/instructions as yet how the minister should record the ceremony.
I have seen (a) ceremonies following a private baptism appearing as a normal baptism (b) the entry in the register indicates "previously baptised/received" or similar (c) no record of the "receiving".
In some places (I've seen evidence of it in Lincolnshire) private baptisms were more or less the norm, and when (or if) the child was later received into the church, this was recorded as a Christening. However, this was a non-standard use of the terms, since the form of words in the Book of Common Prayer only makes sense if "baptism" and "christening" are taken as alternative terms for exactly the same thing.
What I don't think anyone has mentioned is conditional baptism: if there's some doubt as to whether an earlier baptism was performed correctly, the priest performs the ceremony, but with the words "If you have not already been baptised, I baptise you...." Ideally this would go down in the register as a conditional baptism, but I expect it wasn't always.