Our company has an extremely ethinically-diverse clientele. We've got legitimate issues sometimes not being able to recognize which is the man and which is the lady. This is important information so that we can respectfully address them in writing and on the phone. For this reason only, all our internal files are set up uniformly with the protocol of having the man's name listed first and the lady's name second. The customers never, ever see these records but everybody at every level internally can be confident when reading them. As I understand it (it's not my department) our Sales Contracts list whichever person the customer's specify first.
If there's a single person, we put the name in either the first position, if it's a man, or the second position if it's a lady. In our context, there's no reason to list children or any other household members.
As for same-sex couples, well, what can you do? We just pick a name and put it first. Honestly though, this is basically never the case with our ethnic clientele so it's generally easy enough to recognize there are two men's names or two women's.
This is the part that I found weird as well. That and not listing a surname for the OP. Regardless of the whole "head of household" thing, it would seem prudent to list both first-and-last, for every patient.