Jewishness is tricky to define because it legitimately has pluralistic definitions. Under mishpat ivri (secular law), there's one definition for the purposes of aliyah (for example). Under chok ivri (religious law), it has another series of definitions depending on the particular religious community (e.g. Reform doesn't require any initiatory rites and consequently has a broader definition than say Orthodox which defines and regulates Jewishness in myriad ways).
Yet Paul postulated a Messianic community composed of natural branches (ethnic Jews) and grafted-in branches (former gentiles) which is distinct from the cut-off, natural branches (non-Messianic ethnic Jews). And those remaining natural branches, the first-century Messianic Jews, ended up assimilating. They were completely overrun by the Christians and disappeared from history.
But was this Paul's doing? In fact, it was the non-Messianic synagogues that banned Messianic Jews from attending:
"They will ban you from the synagogue; indeed, the time is coming when anyone who kills you will suppose that he is serving God," (John 16:2).
By the way, here's the UMJC's position on "who is a Jew":
"By definition, a Messianic Jew is a Jewish person (see definition of Jewish identity in the Policy Manual)..." pg. 5 of UMJC By-Laws.
But this "Policy Manual" is nowhere to found. However, the Kesher Journal is run by the leaders of the UMJC and in the article entitled "The Case for 'Defacto' Conversion" by Jeffrey Feinberg says this:
"The UMJC has yet to define who is a Jew. Each congregation determines for itself who counts as a Jewish member and who does not. Some congregations might count Jews based on patrilineal descent, others based on matrilineal descent, or both."
Here's an example of one such congregation's definition of a Messianic Jew:
"A Messianic Jew is one who is born of a Jewish parent or was converted according to Rabbinic standards and is joined by faith to Yeshua, the Messiah of Israel," FAQ section of Tikvat Israel's website.
photo credit: Marco Bellucci via photopin cc