There is no Thai Embassy in Yogyakarta, Indonesia. The Royal Thai Embassy is located in Jakarta, which is the capital of Indonesia. Yogyakarta residents or those in surrounding regions needing consular services would typically need to visit the Thai Embassy in Jakarta or use its consular services via their online platform.
is correct that you should be careful staying too many days on visa exempt; however, in my opinion, your previous visits and upcoming visit are far enough apart and within immigration rules, but do be aware that immigration has the righ to question and detain. But you should be ok for the next trip.
do you not understand that the person posting the question has a UK passport. His requirements are different than you as a U.S. passport holder. In your text about needing a Vietnam visa, nowhere did you qualify that this is a U.S. passport requirement.
Just because you needed a visa does not mean everyone from different countries need a visa.