I feel differently on the "Person who practices the Jewish faith", mainly because to me "Jewish" is primarily a cultural thing, and the religion is a closely-related, but different issue, so if you mention the religion explicitly that makes a difference.

That's relevant in some contexts, but tangential to the point I was making above.

Whereas you don't need to do the same for other religions, because they are primarily religions and cultures second.

Seems like you're jumping to conclusions here. All religions are cultural, and it's not so easy to pick the two apart. There are a lot of contradictions in the phrase "I'm just culturally Jewish" (or any other religious designator).