I'll take a stab at it. Around here (AZ) the process is like this. First, obtain a Marriage License from the courthouse. Next, take it to a Chapel and get married. The officiating Minister and witnesses sign the license. Last, take the filled out license back to the courthouse to file. I assume it's that last step the dear Congress rat didn't do.
More about how registration happens:
1. You don't have to register the marriage. If you don't register it, you are not legally married
by law and can't take advantage of any marriage benefits you would have if legally married which means recorded at the county courthouse.
2. If married by a church minster, he may register the marriage at the courthouse for the couple - or not, if the couple does it themselves. They are
legally married according to church teachings, but
not legally married by state law until registered at the courthouse.
3. A very large Muslim mosque is in the greater Dallas area. They marry their people and consider them legally married by their faith. They live according to Sharia law but if they want legal benefits of marriage, they still have to register the marriage at the courthouse.
Basically, you are married in your mind, when both of you want to be married to each other. That is when the marriage starts. The rest is window dressing.