I came to the US on a K-1 visa. After I settled in, I applied asap for my SSN card, which I got under my maiden name, following the guides here in VJ.
Then, before the 90 days, I got married to my petitioner. My marriage certificate has my maiden name because the marriage license had it, but after that, I applied for an update of my SSN card because we wanted to file joint taxes (foreseen the AOS process).
In SSA (I went in person), for an update on my SSN card, I presented my marriage certificate with my maiden name, and I was asked if I wanted to change my name, I said yes, and my updated SSN card came under my married name.
I applied to my Adjustment of Status (AOS) and Remove of Conditions (ROC) under my married name.
When I filled out my electronic N-400, I was asked If I wanted a name change but didn't need it, so I marked no.
During the N-400 interview, the USCIS officer asked me again If I wanted a name change, I said no again,
In my experience, the married name doesn't need to appear on your marriage certificate, your spouse can have the name changed in the SSA and go from there.
I hope this helps.
Best regards!
Maritah.