It is time finally for interpreters, Bible Translators, and commentators to cease and desist. Jesus and the Jesus movement (with all its various movement groups) have their roots in Israel, not ‘Judaism’. They were , in the nascent period, predominantly ‘Israelites’, not ‘Jews’; Galileans, not Judeans; Nazoreans, not ‘Christians’. They belonged to the House of Israel, not ‘Christianity’ (p. 148).
Leaving aside the clearly anachronistic use of the term "Christian" or "Christianity". While we continue to use these terms anachronistically we seem to be more aware of the problem than with the terms "Jewish" and "Jew".
Elliott presents that case that in the Second Temple period outsiders (specially the Romans) referred to Israelites as Ioudaioi and that term was eventually adopted by Diaspora Jews as a self designation especially when conversing with non-Israelites although nonetheless infrequently used. What’s more, Elliott explains that the term Ioudaioi maintained a geographical connotation as it expanded to include ethnicity. While at first the term was coined in the Persian period to refer to those who resided in the region of Ioudaia, it later came to be used as a designation by outsides for all those who oriented their lives—politically, ethnically, economically, socially and culturally—around Judea: the Jerusalem and the Temple, and the cult and law as practiced there. Thus, Elliott argues that the term Ioudaioi be translated as "Judeans", not "Jews" since as he states, "The term ‘Jew’ as a translation for Ioudaios does not communicate the connection of the name with the name of the land" (p. 148).
Also important is his discussion of the term Ioudaisimos which is translated "Jewish". Not only does the term rarely appear in Greek (and has not Hebrew or Aramaic equivalent), but also it was not a conventional term of Israel parlance. He opines, "It is ironic and unfortunate that a name occurring so infrequently [Jew, Jewish] in the literature of Israel and the New Testament should have become in modern times the most frequent designation of the children of Israel" (p. 142-43). Furthermore, Elliott argues that it was used in the Maccabean period to designate not a social collectivity but a "Judean way of behavior".
As a Christian, New Testament scholar and teacher I am as guilty as anyone for the lack of precision in my language. I take Elliott's criticism to heart and will attempt to be much more careful in my writing and speaking. Here is my list of terms to work on: