In antiquity, Ioudaios was an ethnic term that designated the people whose homeland was Judaea, who worshiped the God whose temple was in Jerusalem, who observed the ancestral law (of Moses), and who believed themselves to be descended from Abraham through Isaac and Jacob. In other words, it combined together what we think of as geographical, religious and cultural elements. The term was ambiguous in that it could be used in a narrow geographical sense (Judaean as opposed to Galilean) as well as in its more common ethnic sense (Judaeans from Judaea and Galilee).* The Ioudaioi were an ethnic group considered distinctive in part because of their strange and exclusive views about God. There was, however, no separate religious meaning of the word. Those from other ethnicities who wanted to worship the God whose temple was in Jerusalem and adopt the Ioudaios way of life had to convert from one ethnicity to another.
[*I am now inclined to distinguish between ethnic and geographic meanings of the same term instead of dual or nested ethnicity. This would explain Acts 2:5-13 where the Ioudaioi in 2:5 (here an ethnic designator) consisted of Ioudaioi and Cappadocians (2:9; here a geographic designator). Dual or nested ethnicity could be reserved for instances where someone was a citizen of the Greek ethnos of Alexandria and a Ioudaios at the same time.]
We do not have an English word that does justice to the meaning of Ioudaios. "Jew" captures the religious, cultural and sometimes the narrow ethnic aspects of the word, but misses the strong geographical element. Translating Ioudaios by "Jew" also distinguishes the word unjustifiably from other ethnic nouns. "Judaean" by contrast, captures the tight connection between the people and their homeland, but to a modern ear misses the religious and cultural aspects of the ancient term. "Judaean" might help avoid anachronism, although the danger of anachronism will always linger. "Judaean" also lacks continuity with the ongoing tradition of contemporary Judaism.
I do not think either translation is wrong. I lean towards Judaean in academic settings because its very unfamiliarity encourages more careful reflection on what would have been meant by the term in antiquity. On the other hand, I have no desire to be innovative or to follow the latest fad in my use of terminology. The important thing is to explain the semantic range underlying the word behind our English translation. (I can imagine someone concluding, "So Jesus wasn't a Jew after all. He was just someone from Judaea who lived among a bunch of Jews." ...If one Ioudaios is a Judaean, all Ioudaioi are Judaeans.) Ideally, one would take a course on Second Temple "Judaism." Unfortunately, I do not have a handy catch-all alternative to "Judaism."
These are my tentative conclusions. I have used this series of posts to force myself to think through the literature and issues more carefully than I would otherwise do. One advantage of doing this in a public forum is that others can correct my mistakes, push me to consider implications I may have overlooked and encourage me to revise my conclusions as necessary. Please join in.
Posts in this series:
Part 1: On Jews and Judeans, Israelites and Israelis
Part 2: Ioudaios according to Shaye Cohen
Part 3a: Ioudaios according to Philip Esler
Part 3b: Philip Esler Responds to Shaye Cohen
Part 4: Judean vs. Israelite according to John H. Elliott
Part 5a: Ioudaios according to Steve Mason
Part 5b: Ioudaios according to Steve Mason
Part 6: Preliminary Conclusions