Do I have to apply for the DTV from the country where I live or work?

No you can do from almost any country except those who only let residents apply

Everything depends on which embassy you're applying at.

You can apply from any country that allows you to apply. Some countries only allow citizens/residents to apply

Not exactly. The DTV application does not have to be submitted from your country of citizenship or where your employer is based. It must be submitted from outside Thailand, through a Thai embassy or consulate in a country where you are legally staying. Many people apply from a third country where they are temporarily residing, as long as they can prove legal stay there.

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There is even a user manual