I got that green and red round stamp at the Thai Consulate in Los Angeles after I had the marriage license first authenticated from the US state department and then translated into Thai
Took all of that to the local amphur in Prachinburi and registered our marriage in Thailand
Immigration then wanted all of that including our US marriage certificate when we applied for my one year extension based on marriage
I’ve provide every page that is NOT blank two times for extension of stay based on marriage with no issues and don’t forget the TM30 and TM6 that should be stapled in your passport