a river; in Ezra always the Euphrates, named in the fixed title of the Persian province that lay west of it
This is the everyday Aramaic word for a river, and in the Bible almost all of its work happens inside one fixed phrase. Twelve of the thirteen verses that use it belong to the standing name of a Persian administrative region, "Beyond the River," which took in Syria, Phoenicia, and Judah. Seen from the imperial capitals in Mesopotamia, all of that land lay across the water, so the empire named the whole territory by the crossing. It is the close counterpart of the Hebrew word for a river (נָהָר), which the Old Testament likewise uses of the Euphrates without ever naming it.
Because the phrase is a name, the word functions less like a description and more like an address. Officials write to the king as "the men of Beyond the River" (Ezra 4:11), a Persian governor holds the title "governor of Beyond the River" (Ezra 5:3), and building costs are drawn from "the taxes of Beyond the River" (Ezra 6:8). None of that tells you anything about water. It tells you which slice of the empire is in view. The river is never identified, because in that world only one river needed no naming.
Daniel 7:10 is the single verse where the word does nothing but mean a river. There a river of fire pours out from before the throne in Daniel's vision of judgment. No empire, no province, no official titles: just a stream in motion, and all of its force comes from what is flowing in it.