- pert. to being Judean (Jewish), with focus on adherence to Mosaic tradition, Judean;
- one who is Judean (Jewish), with focus on adherence to Mosaic tradition, a Judean
The question of how to translate Ἰουδαῖος is a tricky one. Does it primarily delimit a set of religious and/or idealogical persuasions, and so mean Jew/Jewish? Or is it rather a geographical and/or ethnic marker, and so mean Judean? BDAG's extended definitions of Ἰουδαῖος recognize the difficulty inasmuch as give "Jewish" in parentheses, but in its glosses BDAG clearly prefers Judean to Jew (which it doesn't offer). This is, in my impression, in line with the current standard position among scholars of Second Temple Judaism, Christian origins, and the New Testament.
Even so, the preference for a geographical/ethnic referent cannot quite shake the religious/idealogical connotations. After the second definition, given above, BDAG offers the following:
Since Jerusalem sets the standard for fidelity to Israel's tradition, and since Jerusalem is located in Judea, Ἰ. frequently suggests conformity to Israel's ancestral belief and practice. in turn, the geographical name provided outsiders with a term that applied to all, including followers of Jesus, who practiced customs variously associated with Judea (note the Roman perception Ac 18:15 ['Judeans' at Corinth]; 23:28).
In his book Border Lines: The Partition of Judaeo-Christianity (Philadelphia: University of Pennsylvania Press, 2004), Daniel Boyarin argues that "Judaism," as a religion, didn't exist before it was invented by early Christian heresiologists (beginning with Justin Martyr) as a foil against which "Christianity" could be defined. In other words, "Christianity" ≠ "Judaism," but before there was a "Christianity" there was no "Judaism." In yet other words, whereas today we think of "Jew" [Ἰουδαῖος] as a belonging to the same class as words like "Christian," "Muslim," "Buddhist," "Hindu," etc. (all of these are examples of religious adherents), before the rise of Christianity "Jew" [Ἰουδαῖος] belonged to a different class, other examples of which include "Roman," "Greek," "Persian," "Egyptian," "Asian," "Syrian," etc.
There's a lot of insight here, but we still need to exercise an inordinate amount of caution. Whereas geographical/ethnic markers today don't necessarily connote religious identity (what do "Europeans" or "Asians" or "Africans" think about God/god/the transcendent?), they clearly did in the ancient world. Being "Roman" or "Egyptian" meant something about your religious beliefs in addition to your national/ethnic identity; if you were "Roman" ethnically but not religiously, that was something to comment on (cp. the complaint against Paul and Silas in Acts 16.20–21). For example, note the description of Dositheus in 3 Maccabees 1.3:
But Ptolemy was taken out of harm's way by Dositheus, called the son of Drimylus, a Judean by race [τὸ γένος Ἰουδαῖος] who later changed his customs [μεταβαλὼν τὰ νόμιμα] and became estranged from his ancestral beliefs [τῶν πατρίων δογμάτων ἀπηλλοτριωμένος].
Surely it's significant, given the questions we've been asking about Ἰουδαῖος for over two decades, that Dositheus's status as a Ἰουδαῖος required qualification (it was "by race" only) because he wasn't a Ἰουδαῖος as the term was normally understood. He may have been a Judean, but he was no longer a Jew.