The term began as Hebrews with Abraham. Then Israelites after Jacob was given new name. Then Jews by the time of Queen Esther after Judah, one of the sons of Jacob. At the time of Jesus, it was Jews living in Judea which is what the region was called at that time because it was where the tribe of Judah had settled. David would have been an Israelite because at that time, the kingdoms of Judah and Israel (Samaria) were still united.
If David had lived 500 years later, he would have been called a Jew. So genealogically, yes, he was a Jew because he was descended from Abraham, the father of the Hebrew/Israelite?Jewish people. And Jesus was descended from Abraham through David, which is why he was called the Son of David, the Lion of the Tribe of Judah.
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