The Church of England, under "administration of the sacraments" published instructions in 1701 for publick, private and riper year baptisms. Under "private" it took the opportunity to discourage home baptisms other than when "need shall compel them" see 1b below. This was taken to be that the infant did not have long to live.
A feature of baptism is the "receiving" by the congregation - see 1a below. This occurs naturally when the child is baptised in church. If baptised at home, and the child survives, a follow up appearance in church is required to make good the "receiving" element by the congregation. The follow up in church must not involve water/anointing.
Recording the baptism, whether at home or church, was required by the state (Thomas Cromwell, 1538). There was no laid down procedure (church or state) to record the follow up "receiving".
In practical terms it was useful to identify the home baptism in the register so that the appropriate rite/ceremony could be performed when the child was later presented for "receiving".
As you can well imagine, what happened in practice was another matter.