Hi Jo,
It would depend on the parish priest whether the rite of reception into the church, which was required, was recorded. As The Prayer Book says, that those children who were baptized privately, were truly and validly baptized, and that if such children " do afterwards live", they ought to be brought to the Church as soon as possible to be received as members of " the flock of true Christian people".
I have seen baptism registers where it says "X was privately baptised on ***** and received into the Church on ******"
Some vicars used "Christening" to mean the reception into church rite.
Stan