Note though that Jewish basically means єврейський (related to єврей, a Jew), regardless of the religion itself. the term єврей is neutral (like Jew in English), but іудей is sometimes better to use, like in this sentence, yes.
Does one imply religion and one ethnicity?
іудаїзм = Judaism
іудеї = Jews (practicers of Judaism)
єврейський = Hebrew (adj.) or Jewish (ethnic identity)
єврей = Hebrew (n.) or Jew (ethnic identity)
Again, the issue arises of what is the natural equivalent in English. We would not say 'Jewish Religion': we would say "The jewish / Jewish religion."
"Jewish" is never lower-case