Fun at the consulate: What even is a “certificate of marriage”?

My wife and I, both US citizens, are applying for permanent residency. One of the things on our list of documents is a certificate of marriage. Of course, we have our official certified copy of our notice of vital record from the county (we were married in the US), but apparently we need to get… something from the US consulate.

Our attorney showed us a (redacted) example from a client from Australia, which was just a marriage certificate from the state with a notary seal from the Australian consulate in Japan. Seemed simple enough.

So, I made an appointment at the US consulate, and everything went sideways.

1. They told us we needed to bring a witness, so I managed to find a friend who wasn't working that day and entered all her information on the appointment. During the appointment, they never asked for a witness, but they did threaten to arrest our friend for being in the building without authorization.
2. They insisted that the State Department provides no such service, even though it's listed on the consular services website and it was the option I selected when I booked my appointment.
3. After several hours of rudeness, they had me hand write an affidavit with all of the information in our marriage license, and then they notarized that.

So… what the hell? I have no idea if this random affidavit serves the purpose of a "marriage certificate." My lawyer is flummoxed too.

Does anyone know what the MoJ actually expects for this?

Edit :

Here is what the US State Department website says about this :

https://jp.usembassy.gov/passports/marriage-certificate/

As you can see, it's ambiguous and doesn't actually say that they will do anything.

The MoJ instructions for my particular situation do not actually mention a marriage certificate :

https://www.moj.go.jp/isa/applications/procedures/nyuukokukanri07_00130.html

When I applied for my current visa (HSP), I submitted our notice of vital record (what other states probably call a marriage license) from the state of California, and that was accepted without issue. The only place I've heard of needing to get a certificate from the embassy or consulate is from my lawyer.

USA.gov says that the US Government does not issue marriage certificates, and that they must be obtained from the state where the marriage was performed. This is what we already have, and it worked for getting my wife's dependent visa when I applied for my HSP visa.

https://www.usa.gov/marriage-certificate#:~:text=How%20to%20get%20a%20certified,by%20mail%2C%20or%20in%20person.

To clarify my question, is this "marriage certificate" from the US Embassy a real thing, or is is this just my lawyer being confused about how the United States works?