Second, and as we all know, defining the term "Jewish Christianity" is like nailing jelly to a wall given the nature of our sources (are Matthew and John Jewish Christian writings, if so, which is more Jewish?; is Gospel of the Nazoreans a rip off from Matthew?), the sociological issue of how to know if someone is in or out of the Jewish group, and pluriform views on adherence toTorah by ethnic Jews and ethnic Gentiles in the early church.
Third, Matt Jackson-McCabe writes: "If we designate 'Christ-believing Jews' those Christ-believing Jews who agree with Paul about the primary identity of Christ-believers and the central definition of the covenant community, then we may want to designate those who he opposes in Philippians and 'those from Jews' as Jewish Christ-believers. The latter expression would then indicate that their primary identity is Jewish; that is, they are most fundamentally faithful adherents to the Mosaic covenant. Their belief in Christ, then, functions within that sphere of identity. Our study indicates that there is so single way - or even two ways - of being a Christ-believer and Jewish in the first century. However Christ-believers understood the mission and work of Christ in relation to Israel and Gentiles varied significantly. There is enough variation among Christ-believing Jews that a single designation for them is misleading, particularly if that label's central function is to distinguish them from Paul or from the Pauline churches. Serious theological issues divided Paul from some other Christ-believers, but some times those on his own side would have been Jews and some times Gentiles and probably nearly always a mixture of the two" (pp. 77-78).
Fourth, I am not a fan of "Christian Judaism" since that sounds like Jewish Muslim or Liberal Evangelical. I think the title "Judeo-Christianity" is a better way of designating Christ-believing Jews who still find their identity (and therefore their praxis) as bound up with the Torah/Covenant while allowing for variation of how that worked itself out in reality.