Lost Tribes of Judah or Israel?

The core element of the Jewish People are descendants of inhabitants of the Kingdom of Judah. The Ten Tribes had separated from Judah and set up their own Kingdom of Israel. The dwellers of this kingdom were exiled by Assyria and became lost. The Kingdom of Judah remained. It consisted of the Tribes of Judah, Benjamin, and Levi, along with representative minorities from the other Tribes. The term Jew is derived from the name "Judah." They did not lose consciousness of their identity. They remained faithful in a basic sense to the Torah Law i.e. to Judaism. The term "Judah" in Biblical prophecy is applied to those who belonged to the Kingdom of Judah. We thus have Mordecai uncle of Esther who was from the Tribe of Benjamin but is referred to as a "Ish Yehudi" i.e. "a Jewish man".

Esther (NASB) 2:
 5 Now there was at the citadel in Susa a Jew whose name was Mordecai, the son of Jair, the son of Shimei, the son of Kish, a Benjamite, 6 who had been taken into exile from Jerusalem with the captives who had been exiled with Jeconiah king of Judah, whom Nebuchadnezzar the king of Babylon had exiled. 7 He was bringing up Hadassah, that is Esther, his uncle's daughter, for she had no father or mother. Now the young lady was beautiful of form and face, and when her father and her mother died, Mordecai took her as his own daughter.

The Lost Ten Tribes are sometimes termed "The Ten Lost Jewish Tribes." This is mistaken but understandable. For more details see our answer to the related query:

Names of LTTs. What Israelite tribes are the considered lost?