Right.
> We get the word "Jew" from Judah.
Only in the sense that the word "Jew" eventually came from the OFFSPRING of the TRIBEs of Judah and Benjamin and Levi which formed the Southern Kingdom, but over a Thousand Years after Judah himself died. See my Profile for more...
But David was not a Jew. "Jews" didn't appear until almost 500 years after Davids death, when they returned from Babylon. Davids Kingdom of Israel contained all 12 tribes, including the Tribe of Judah, which was not "Jewish".