Could not get to speak to a person at New York or Los Angeles. I even tried calling the Home Office in Croydon. Couldn't get through either.
With Chicago, a human being actually picked up the phone and asked me if I wanted to be put on hold. The person who answered my question was extremely courteous, even apologized for having to put me on hold a few times to speak with her supervisor.
For anyone who is interested, I was told that I do not have to report a same-sex domestic partnership with a fiance visa application, because it is not a legal marriage. They are only interested in legal marriages and divorces. I did give a brief explanation of what the dp is, and explained that it was legally valid in New York. In fact, from her tone of voice, it seemed like she would be annoyed if she were given an unnecessary piece of paper. (Note, this information only applies to a fiance visa; I didn't ask about the other types of visas. Also, she specifically asked me, after speaking to her supervisor, if my previous relationship was same-sex. So the rules for a previous opposite sex common law/domestic partnership-type relationship might be different. Unless you are in the exact same situation as me, you should call the Consulate directly if you have a similar question.)
P.S. I said hurray because it was so easy to get through to them and they were so nice. Not because of the information that I was given. I would have accepted whatever advice they gave.
I also wanted to recommend that anybody who has a question about anything complicated to try calling Chicago first. They answer inquiries starting 2PM Central Time.
P.P.S. I had previously checked the Home Office Guidance Notes regarding impediments to marriage. Basically, if the applicant is an adult, they are directed to assume that a verbal statement made about his or her marital status is truthful. Unless they have good cause to be suspicious, in which case the officer is to make the appropriate investigation.