It's interesting that in Ezra 5-6 the Persian kings are once referred to as Babylonian and once even as Assyrian. Any thoughts on this? 
  • However, in the first year of Cyrus king of Babylon, King Cyrus issued a decree to rebuild this house of God (5:13).
  • And they observed the Feast of Unleavened Bread seven days with joy, for the Lord had caused them to rejoice, and had turned the heart of the king of Assyria toward them to  encourage them in the work of the house of God, the God of Israel (6:22).

It seems like sloppy labeling. Of course there's a lot of geographic overlap between Assyria and Babylon, but anyone knowledgeable would be hard-pressed to associate Assyria with anything positive -- even those from Judah. -- S.M.J.

Good question. I noticed an apparent confusion of geographical terms in the O.T. Apocrypha (and even used this as evidence to minimize the importance of the Apocrypha -- unfairly, I think), but you have identified a clear example in the canonical text. (Thanks.) See also Chron 33:11: "Therefore the LORD brought against them the commanders of the army of the king of Assyria [the empire], who took Manasseh captive in manacles, bound him with fetters, and brought him to Babylon [the city]..." Babylon is a city within the land of Mesopotamia / Sumeria / Assyria / Babylon -- and, from 539 BC, the Persian Empire.
I don't think the terms have to be used strictly. Babylon is political -- a city. And imperial -- an empire. And geographical -- a land (Babylonia). Since Assyria and Babylon occupied roughly the same territory (more or less congruent with modern Iraq), the terms could be used interchangeably. Unless Ezra is a work of geography calling us to precision, I doubt the discrepancy is problematic.
This may also be a backward glance at all the pain and suffering that the Jewish people had been suffering (since the 8th century BC). They either lived in their land (the Promised Land of Israel) or they were in foreign / enemy / pagan / idolatrous territory. Assyrians and Babylonians had a lot in common with each other -- far more than with the Israelites.