I was using the term Israelite to denote descendants of the patriarch Jacob, who was named Israel by God. But you are correct that the Babylonian exile occurred after the Canaan “promised land” was divided into the two kingdoms, Israel and Judea (from which the word "Jew" arose). Both kingdoms contained the same Hebrew-speaking semitic ethnicity, did they not? Judea was mainly the descendants of Jacob's son Judah, and the eleven other tribes lived mainly in the Israel kingdom, if I understand correctly.
As for debate, the Hebrew Bible makes reference to the long geneaologies of its characters; and the Book of Nehemiah describes Nehemiah's and Ezra's efforts to identify the returning Jews by or with their parentage, and to extinguish relations with non-Jews.
I’ve never seen it mentioned on FR that Judea had conquered and assimilated the Edomites prior to their Babylonian exile.