Correct, he lived in Turkey around the late third-early fourth century.

Pretty sure he lived in Anatolia or the Roman empire. There was no 'Turkey' to live in around the late third century.

He can't have lived in the Roman Empire, because those are words written in English, a language that didn't exist back then.

What's that? You meant, he lived within the bounds of the region that we call one thing, but would have been something else contemporaneously, but both refer to the same geographical location? Great, we agree he lived in Turkey.