beancounter, I don't think you're getting the part that's confusing me.Think of it this way. You can't change your ethnic background or your race, but you can change your religion whenever you want.
Judiasm is a religion, because you can become part of it or leave it on a whim.
Also, I added a couple words to you statement (in red) that I think should help clarify things.
Examples:
Scottish = ethnicity, descent, and/or race
Protestant = religious affiliation
vs.
Jewish = ethnicity, descent, and/or race (but only through the mothers' line - a rule that doesn't apply to any other line of ethnicity, descent, and/or race that I know of)
Jewish = religious affiliation (unless you're not a follower of Judaism, in which case it means the above)
So if you're Jewish either your mother was Jewish, or you've converted to Judaism. It's not a very precise term as it can mean more than one thing.