Just got into Phuket yesterday. If admin is interested in me sharing, here's my experience.
Gate agent in the US checked PCR test (highlighted key fields). She noted concerns and had to call a manager because the test said "Rapid" somewhere on it--even though it was PCR. Ended up letting us pass without incident, but should be aware not all agents may operate the same.
Gate agent at transit port (last one before HKT) checked CoE (not PCR test)
Landed in HKT, disembarked, and everyone sat down in spaced-out chairs. Health Control came by and checked 4 documents: Arrival Card, CoE, Passport and pre-flight PCR test. (these were the only documents used to get through the process)
After they were prechecked, you queued up to go by a desk where the documents were verified and put into the system. After this, they kept the PCR test (no issues) and handed the others back. (I applied under Sandbox scheme, but approved as Test & Go)
You then take the 3 documents through Passport Control. They'll keep the Arrival card (don't lose the departure card; keep until you leave Thailand) and you can now put CoE away.
After Passport control, you get your bags and move through customs.
After you have your bags, you'll proceed to leave and go through the COVID testing area. Present only your passport, and they'll give you a testing kit. Take that to the testing window.
Then, find your driver and head to your hotel. After getting to your room, they'll have you check in with MorChana and ask you to stay in until the PCR results come back. Ours took 4-5 hours and now we're free to move!