Messianic Jews and Christians would disagree with you. God decides who is Jewish not men. Born a Jew always a Jew. As Tucker said they are completed Jews, as is Jesus Who is still Jewish.
What is the difference between a Messianic Jew and a Christian ? I submit that there is no difference. Therefore, to use both of those terms as if they signify two kinds of people is confusing. There are Jews and Gentiles, male and female, bond and free who comprise one body in the Messiah.