Interestingly, I've gone the opposite way. "Judeo-Christian" is a phrase I've gotten tetchy about as I've gotten older. I think it has to do with learning the history of the word. There's mixed responses to the validity of the word among Jews, but these days it's so broadly used among philo-semites and Christian end-times Zionists that it feels icky.

(Also co-sign to the responses to your initial worry about Satan, alexseanchai. Good instinct.)