You're partially correct, but there's more.... Perhaps, you haven't investigated the origin of the name 'Jew' and how it has been completely mis-characterized. Word manipulation and name-stealing has a long history. The word 'Jew' derives actually from the French 'Juerie'. I will get to the significance of this in a moment. First let me begin with the history of Judah, as a living, breathing individual and the land of his posterity, which became known as Judah, until his posterity was forcibly and completely removed from the land of Judah to Babylonia. The proper word describing Judah's offspring and descendants has always been Judahites. The word 'Jew' never derived from Judah and certainly did not exist during the Roman times as I will show in the following paragraph.

Rome referred to the region in the Latin as 'Iudaea' (Engl.: 'Judea'), which was formerly called by the Greeks 'Idoumaia'; which means the "land of Edomites". Since the Greco-Roman were deeply intertwined over the centuries, the Romans heavily benefited from the Greek influence in areas of trade, banking, administration, art, literature, philosophy and earth science. Not only did the Romans adopt Greek influences, it Latinized Greek place names. The Greek influence on Roman names can be seen by the Latin name of Egypt, which is 'Aegyptus' deriving from the ancient Greek 'Aigyptos'. The ancient Egyptians didn't call their land either the ancient Greek or Latin place names, but ' Hut-Ka-Ptah'.

Similarly, the Judahites did not refer to themselves by either the ancient Greek or Latin name. Alexander the Great conquered the region and called the land Idoumea after the predominant inhabitants of this region. The English name John is shown here as another example of a Latinized name deriving from the ancient Greek. The Greek name Ἰωάννης became Johannes in Latin and then John in English, but in modern Greek it has become Γιάννης; this might be written as Yannis, Jani, Ioannis, Yiannis, or Giannis, but not Giannes or Giannēs as it would be for ancient Greek. https://en.wikipedia.org/wiki/Romanization_of_Greek

The influence of the Greeks in the entire Levant was so culturally dominating that the Hebrew written language itself was threatened. This written language is referred to as paleo-Hebrew and it has been lost in antiquity for more than 2000 years. Thanks to Ptolemy Philaphelphus and the 70 Isrealite scholars he assembled, the Greek Septuagint was written in 256 BC and as a result, saved the Old Testament from being lost in antiquity. The international language throughout the eastern Mediterranean was koine Greek. Without records, it is plausible that Jesus may have read from the Greek Septuagint. We don't know for sure. Nevertheless, the Romans readily accepted the Greek influence existing there and added their own Latin version. The Romans simply used the Latinized version ('Iudaea') of the Greek 'Idumea'. 'Iudaea' in English is Judea.

Much later the word Jew comes into existence in England in circa 1600s, which coincides with a wave of Yiddish immigrants coming from France and Deutschland. These Yiddish settlers came from eastern Europe and originated from Khazaria, not the Middle East, but rather the steppes of Caspian and Black Seas, which had since fallen to the proxy Byzantine and Caliphate conquering armies. The French and Europeans viewed these immigrants negatively and treated them similarly to gypsies. They allowed them to quarter in only a designated area of the city. A French derogatory term for 'ghetto' and the Yiddish district of town was called – 'Jeuerie'; "ghetto", from Anglo-French 'Juerie', Old French 'Juierie’ or the later English version 'Jewry'. Originally the English term 'Jewry' referred to those immigrants coming from Eastern European people who spoke Yiddish (Ashkenazi). The word 'Jew' did not exist during the Roman times.

In 2001, the third edition of the Bauer lexicon, one of the most highly respected dictionaries of Biblical Greek,[12] supported translation of the term as "Judean", writing. Here is what it concludes:

Incalculable harm has been caused by simply glossing Ioudaios with ‘Jew,’ for many readers or auditors of Bible translations do not practice the historical judgment necessary to distinguish between circumstances and events of an ancient time and contemporary ethnic-religious-social realities, with the result that anti-Judaism in the modern sense of the term is needlessly fostered through biblical texts.

It conveniently leaves out any words of 'intentional' word manipulation or 'name-stealing'. In this regard, we are also left to believe the papacy arising out of the Roman era didn't manipulate Roman pagan holidays, including their names, to coincide with important Christian events. Well, we know this is in fact true. Why then is it ignored in other instances?

Academic publications in the last ten to fifteen years increasingly use the term Judeans rather than Jews. [There is no distinction](See https://web.archive.org/web/20120720012434/https://en.wikipedia.org/wiki/Ioudaios) between "Judahites," "Judeans" and "Jews" in modern standard Arabic. First appearance in Wikipedia, July 12, 2012. The word 'Jew' has been intermingled with the words 'Israelite', 'Judahite', 'Judean', 'pharisee', and 'Edomite' for over 400 years. It's use in the Bible is ambiguous and unfortunately needs to be deciphered for every verse that it is used. Some times it refers to the Roman province of Judea, other times it refers more accurately to Edomites. Other times it references 'Israelites". Fortunately, some scholars are starting to take notice.