I started Shaye Cohen's The Beginnings of Jewishness: Boundaries, Varieties, Uncertainties. The book is about what in antiquity marked a person as a Jew. Cohen offers documentation that Gentiles did not identify Jews as Jews on the basis of their appearance or clothing, or even their circumcision, for Gentiles could not see that on Jews in their day-to-day lives (though Cohen does mention a story in which Joseph reveals his identity to his brothers by showing them he was circumcised). Rather, Gentiles identified Jews as Jews because the Jews primarily associated with other Jews, observed certain customs, and worshiped the God of Israel.
Then there was Herod, who was also relevant to the question of "Who is a Jew?" According to Josephus, Herod was half-Idumean, and Idumea was technically a part of Judah and was forcibly converted to Judaism a few centuries before. But Herod's mother was Arab. And so Herod was a Judean through his father, meaning that the custom that one is Jewish through his mother had not yet come into play. But Herod did not believe that his Judean credentials were strong enough, and so he commissioned his historian to make him look more Judean---as in, descended from the exiles who returned from Babylon (which Cohen equates to the pilgrims arriving to America on the Mayflower). And then there were people who did not believe that Herod even was Judean.
Cohen also talks about the Greek word "Ioudaios." He thinks that it means Judean---in a geographical and a cultural sense---and yet Cohen mentions a lot of evidence that one could be a Judean in the exile, away from the land of Judea. "Jew", however, which it later meant, was more of a religious term, in that it refers to the Jewish religion. So there is a difference between "Judean" and "Jew", as far as Cohen is concerned, and I may have been misusing the terms in this post. But (for me as I write this post), it's late.