If I convert to Judaism, I am indeed a Jew. If you are referring to being a Hebrew, however, I can see your meaning. But, then you also have the term Israeli, which mixes more people in. And, considering the original test of being a Hebrew was, being from a Hebrew womb, you isolate more people. Add in 4000 years of outside marriages and childbirths, and I believe very few people know if they are truly a 'Chosen One'.
Today, the various Jewish denominations have sharp disagreements about the nature of the Messiah and the Messianic Age, with some groups holding that the Messiah will be a person and other groups holding that the Messiah is a representation of the Messianic Age itself. Traditional thought and current Orthodox thought has mainly held that the Messiah will be an anointed one (messiah) descended from his father through the Davidic line of King David who will gather the Jews back in the Holy Land, the Land of Israel, and usher in an era of peace. Other denominations, such as Reform Judaism, perceive a Messianic Age when the world will be at peace, but do not agree if there will be a messiah as the leader of this era.
And a jew for jesus is no longer a jew, simple. In fact, Israel has refused citizenship to many JfJ as they no longer are classified Jewish.