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Syria is contiguous with Mesopotamia.
The area to the south of Syria but east of Palestina and the five kingdoms (Israel, Judah, Ammon, Moab, Edom) is Arabia. In ancient times the eastern desert was sometimes generally called Ammon.
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Historically, Syria and Assyria are the same. “Syria” is a Greek version of “Assyria”. In ancient times “Syria” designated the whole of the Assyrian empire. The Christians in Iraq call themselves Suryāyē “Syrians”. The modern country of Syria got its name in the period of French rule.
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the area of northern Mesopotamia and eastern Syria is also known as the Jazira (or Al-Jazira) but this does not include the Levant and the Damascus and Aleppo regions.
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Wikipedia seems to be saying that that entire area up to the Euphrates is considered part of Syria (or "Greater Syria" if you prefer). You are correct that the English name for this particular desert (and only that desert), is "The Syrian Desert".
Historically anywhere that is mostly uninhabited is going to have fairly vague political boundaries. If there's almost nobody living there, its going to be a waste of manpower to send armies there to enforce your political will on what few inhabitants there are, and most governmental amenities would be prohibitive to provide. It is possession of the bits on either side that matters, not the desert in-between. In many cases, its nice to have a bit of a buffer area between the two.
The Bible tended to refer to any area that wasn't inhabited by farming peoples as "the wilderness" (depending on your translation).