Delivery (childbirth) at home, at a home, in a home?

"at home" means one's own home. If, for example, you were visiting your parents and gave birth in their home, you didn't give birth "at home" but rather "in a home."

And if you are debating now between "in a home" vs. "at a home" because of the circumstance in which you gave birth outside but still on the premises, just know that "at a home," while not ungrammatical, is very bizarre and unnatural phrasing. It's a construct someone not fluent in English would use. When we hear "at a home" we expect there to be more to the phrase: "at a home for the deaf," "at a home my mother was renting out." A commenter above, BoldBen, is spot-on when he points out: 'At a home' implies somewhere other than the patient's own home and suggests an institution of some kind, either a care home, residential home or, if there are any left, maternity home.

The phrasing of "in a home" is much, much more natural.