I am looking for the appropriate name(s) for people living in the Roman province of Judea during the first century CE who were ethnically, culturally, and religiously Jewish.
What name(s) did they use for themselves? What name(s) did the Romans have for them? What other names were associated with them? (preferably in Latin/Aramaic/Hebrew or other regional languages at that time)
Historians generally call them "Judeans." The word "Judaioi," which is what they were called at the time in Greek, eventually came to mean "Jews," so some historians say "Jews" for the first century too. This is supported by the fact that they used the same word for people whose ancestors were from Judea but who were living in, say, Alexandria, or somewhere else in the empire. But if you're talking about the ones who live in Judea, Judean is probably a safer translation.
The people themselves, when they were speaking Hebrew, generally used "Yisrael" (Israel).