This is based on information in "The Complete Gospels", Robert J. Miller, Editor, Harper Collins, 1992.
"Now Pilate wrote a title and put it on the cross... JESUS OF NAZARETH, THE KING OF THE JEWS... and it was written in Hebrew, Greek, and Latin."
--Gsp. John 19:19-20, likely written between the last decade of the first century, C.E. and the third decade of the second century, after the fall of Jerusalem to Roman legions at the end of the second Jewish war against Roman occupation.
Answering the questions requires some background. The religion/culture we call Judaism scholars tend to locate in three historical eras.
1. First temple religion, c. 950-586 B.C.E.
This is the era of the "Israelites", in which King Solomon's temple was built. The Babylonians sacked, then brought it down, when they conquered the Southern Kingdom, Judah.
So: Israel, Israelites, Torah-based religious culture.
2. Second temple religion, c. 520 B.C.E. -70 C.E.
This is the era of the "Judeans'" temple which, according to several gospels, Jesus and his companions visited at the Passover Festival when Jesus was most likely thirty-three, days before his murder. Note again that people of this second temple era in and around Jerusalem were known as "Judeans", that is, people living in Judah. (They were not "Jews"; bear this in mind as we move forward.)
So: Judea, Judeans, Torah-based religious culture.
3. Rabbinic/Synagogue religion, c. 90 C.E. onward
When, in 70 C.E., Roman legions destroyed the second temple to end what Jewish historian and Roman sympathizer, Josephus, called the (second) 'Jewish War', Judaism forever lost its central, high priest/animal sacrifice religious paradigm. What we know now as Judaism is what survived, then developed and thrived (in Judea/Israel and then, over time, worldwide) without a central worship site and hierarchy/authority.
So: Jews, Judaism, rabbinic/synagogue, Torah (and then Torah- and Talmud-based*) religious culture. "Jews", is what makes sense to call adherents of the religion and culture of that post-70 C.E. survival and eventual thriving and geographical diversification.
So, why does it matter?
When the gospels, written in Greek, tell us that Romans placed a sign on the cross above Jesus' head, that sign, at the time of Jesus' execution, could not have said
KING OF THE JEWS.
"Jews" was not a term used by Jesus' earliest companions and followers, nor would it have been a term Rome would have used. If there was such a sign placed over Jesus' head, it would have read
KING OF THE JUDEANS.
When the first written gospels were copied, redacted, re-copied and orally spread, and later widely distributed in written form after the second temple's destruction in 70 C.E., the term "the Jews" increasingly would have supplanted "the Judeans".
Why does this matter? It matters because
1. Jesus' first followers identified with the religion and culture of the second temple, the Judean culture's religion.
2. The developing schism between the Judeans and the Judean-Christians, who, by the time of John's gospel in the early second century...that growing schism resulted in Christian polemics that saw an increasing intensification and calcification of an ideology that said that, for not accepting the new religion, Jews (not "the Judeans") are liable for punishment in the crime of Jesus' murder.
3. Whoever wrote the gospels, no colonized and exploited person with a brain blames in writing his brutal colonizers for the murder of any increasingly adored religio-cultural icon.
Most important is this:
4a. "Judeans" speaks to a religious culture located in a specific time and place.
b. "Jews" refers to every religious and cultural descendent of that original Judean religious culture.
c. "Judeans" cannot be thought, now, to be continually liable for punishment in the matter of Jesus' murder.
d. "Jews" can be, however inaccurately, identified as continually liable for punishment in the matter of Jesus' murder because of how late first/second-century C.E. gospel and subsequent religious language influenced, if not altered, history.
Professor Miller (see above), of Juniata College in Pennsylvania, concludes:
"...the stereotyped opponents of Jesus in the Gospel of John" ought to be called not Jews but Judeans.
I do not believe that in my lifetime that any language changes among scholars will have a serious, ameliorative effect on people's attitudes. I'm not naive about the weight of history nor about how awful people can be. Nor am I pollyannish about how resistant bigotry is. At the same time, I do not believe that the problem of Christian anti-Semitism is, now, quite the horror it has been.
And yet a conscious Jew is an alert Jew.
Nonetheless, as a matter of historical accuracy, I think scholars changing "the Jews" to "the Judeans" makes sense. Too, if it can over generations make a difference in human relations, it's at the least a good stab. I applaud Professor Miller and his colleagues and their more accurate translations.
* Talmud is the vast collection of medieval rabbinical writings and disputations derived from Torah law.