The Book of Common Prayer has two forms of service, Private Baptism... in houses and Publick Baptism... in the church. Although we tend to view the latter as a small family occasion, properly it seems to have been conducted during a regular service, after the second lesson. Therefore the child was received into the congregation at baptism.
The private baptism service is very short, three prayers and "I baptise thee... ". It is followed by a form of words to be used when the child was first brought to church. The minister certified the time and place of the private baptism and the godparents gave the name of the child, who was then received into the congregation - plus prayers etc.
Amended