You are correct....plus the fact that it is no where mentioned in scripture that they returned.....while two entire books (Ezra & Nehemiah) are devoted to the return from Babylon of the Jews about 525 B.C. This would have been about 200 years after the Assyrian exile of Israel.
It is after the Babylonian captivity that the Books of the Kings and the Chronicles are written and the Southern Kingdom is hereafter referred to as Jewish. The Northern Kingdom continues to just be called Israel.
Remember.....during the first century Josephus records that the people of the Ten Tribes, beyond the Euphrates, are so numerous....they cannot be counted!
The first occurance of the name "Jew" is in 2 Kings.