This is like telling a Jew they're a Christian because the old testament is heavily cribbed from Jewish texts.

Not quite. It's like telling someone who says they're Jewish, that Christ was the true savior and son of God, and that he brought a new covenant that comes before the covenant formed with moses that they're describing Christianity.

That's exaggerating the point, but my meaning is that while your beliefs are your's and it's rude to correct someone on them, the word used to describe those beliefs can be much more readily discussed.