> And I'm not aware of either term...being used to apply to the period from the 8th - 13th centuries
The Journal of Near Eastern Studies (https://www.jstor.org/journal/jneareaststud) is typical in defining its scope as the Near East, from the ancient times to pre-modern Near East. As I said, "Near East" is the standard term for referring to the history/archaeology of this cultural region in pre-modern times.
The Journal of Ancient Near Eastern History seeks to encourage and stimulate the study of the history of the ancient Near East, which is broadly defined to include areas from Iran to the western Anatolian coast and the Black Sea to Southern Arabia from its prehistoric foundations to the Late Antique period.
The Late Antique period is very roughly the 3rd-7th centuries.
The Journal of Near Eastern Studies (https://www.jstor.org/journal/jneareaststud) is typical in defining its scope as the Near East, from the ancient times to pre-modern Near East. As I said, "Near East" is the standard term for referring to the history/archaeology of this cultural region in pre-modern times.
When you want to specify the pre-Islamic Near East, the standard term is "Ancient Near East", as this from the Journal of Ancient Near Eastern History (https://www.degruyter.com/journal/key/janeh/html?lang=en).
The Journal of Ancient Near Eastern History seeks to encourage and stimulate the study of the history of the ancient Near East, which is broadly defined to include areas from Iran to the western Anatolian coast and the Black Sea to Southern Arabia from its prehistoric foundations to the Late Antique period.
The Late Antique period is very roughly the 3rd-7th centuries.