Thanks. As an EU citizen am I exempt from this order? How does the judge or the NYTimes or OpenAI know that I am an EU citizen?
The court in question has no obligations to you at all.
The current legal stance in the US seems to be that you, not being a US person, have no particular legally protected interest in privacy at all, so you have nothing to complain about here and can’t even sue. The only avenue the EU would have to change that is the diplomatic one, but the Commission does not seem to care.