Upvote:1
For immigration you should use the same passport as you used when you entered the country. They are likely to check your entering stamp when you leave the country to see if you have overstayed. For more on travelling with two passports, this answer has some useful answers.
If the name in your ticket and passport match at least partly, you should be fine. There are many different conventions to writing names around the world.
Upvote:10
Immigration and airline check in are two totally separate procedures and systems.
You entered Thailand on your Mexico passport, you need to exit Thailand on your Mexico passport. You can not exit using another passport, as the data on your immigration forms (and thus the data on the terminal of the Immigration officer dealing with you) will have your Mexico passport.
Checking in for your flight involves which ever passport you plan to enter Korea on. It doesn't matter which you use, as long as it is valid for your entry into the destination. The airline may request to see both passports, since your US passport would not have any Thailand entry stamps.
My daughter is a dualie and we usually use her US passport for check in and her Thai passport for Immigration. No hassles, but often have to show her Thai passport at check in as well.