...or perhaps more accurately, why I refuse to answer to the above description.
Generally I'm not very fussy about the terminology. I'm happy to call myself a Messianic Jew, but am okay about being called a Jewish Christian. After all, the literal meanings of Messianic and Christian are identical - though of course there are centuries of baggage attached to the word Christian, which is why I fully understand why some of my fellow Messianic Jews are not comfortable using that term.
I'm also comfortable with talking about Jewish Believers within the body of Believers, but once you use this term with other people it can be confusing - believer in what/who? (I have also come across the term JBY - Jewish Believer in Yeshua - which is more precise, but also a bit of a mouthful...)
But I was miffed when recently I heard someone talk about "Christians of Jewish Background", because to me this terminology seems to imply that our Jewishness is something that we shed when we come to faith in Messiah Yeshua, which is ridiculous especially considering that he is Jewish himself, and he never came to start a new religion!
P.S. For those unfamiliar with the Hebrew terminology, I'd better clarify: Yeshua is the Hebrew name which has been translated into English as Jesus. Messiah is translated into English (through Greek) as Christ.