"Jew" was commonly used to identify this nation of people.
I gave ample evidence in my post as to who the title Jew began to be applied to, i.e., those who returned from the Babylon captivity to Jerusalem/Judea. Ezra 2 mentions NONE of the ten tribes returning, and even in Ezra 8 it shows Ezra had to send the head Nethinim (temple servant) back to Babylon to get Levites because Ezra saw none among the people returning.
The title of Jew comes from the sole tribe of Judah, and it applied ONLY to the southern "kingdom of Judah" that went captive to Babylon for 70 years.
But of course modern Jewish leaders like to throw that title Jew around wrongly applying it to
all Israelites. The reason they get by with that is because they know the majority aren't really going to study their Bible history.
1. God split old Israel into two kingdoms
a. Rehoboam of Judah, son of Solomon, over tribes of Judah, Benjamin, and eventually Levi, in the south at Jerusalem, called the "kingdom of JUDAH".
b. Jeroboam of the tribe of Ephraim, God gave him to be king over ten tribes in the north at Samaria, called the "kingdom of ISRAEL".
2. both kingdoms had war against each other, Rehoboam trying to bring back the whole kingdom.
3. Jeroboam setup idol worship in the north. The Levites left and went south and joined with Judah. Some small remnants of the ten tribes also went south to Jerusalem. The majority of the ten tribes remained in the north with Jeroboam.
4. God brought the kings of Assyria upon the ten northern tribe kingdom and ended it, removing the ten tribes to Assyria and the land of the Medes, never to return, even to this day.
5. ONLY the "house of Judah", or "kingdom of Judah" remained in the land, south at Jerusalem-Judea. They were made up of the tribes of Judah, Benjamin, and Levi.
6. About 120 years after God had removed the ten tribes, the "house of Judah" also fell away to false worship. God then brought the king of Babylon, Nebuchadnezzar upon Jerusalem, and took them captive to Babylon for 70 years. Only a small remnant of the "house of Judah" returned to Jerusalem after the 70 years.
That final state of the "house of Judah" is who used the title of JEW for themselves. That became a JEWISH state. The majority of the ten tribes were gone, and were NOT part of it.