I just looked it up because of curiosity in knowing that the current King of Thailand was born in the USA. Because his parents were here and were part of the royal family, they did not fall under the jurisdiction of the US government, thus, the King of Thailand is not a USA citizen even though he was born here.
Someone should mention to the NYT that the King of Thailand could run for USA presidency. If they can get one thing wrong without bothering to check, I would imagine they would gladly run a story about a foriegn monarch being eligible to run for US President. Plus, there is the whole other thing about residing for a certain amount of time in the USA to be eligible to be President.
You are absolutely correct. Also, the children of foreign diplomats born in the US are not US citizens.