Incorrect and too simplified.
At time of Christ, Palestine was a Roman Province named "Judea". The Hebrews of the Bible became the founding people of both "Israel" divided into twelve tribal names. Judea was one of them, in the South.
100 yrs before Christ, another ppl further to the South, the Edomites, infiltrated and subverted the nations of Israel and Judea, primarily via the judiciary (Saducees) and legal class (Pharisees).
King Herod, who murdered all the babies looking for Jesus, was an Edomite by ethnicity, not an Israelite/Judean.
Even modern Jews acknowledge in their own writings that they are descended from Edom, not Jacob/Israel.
But just as immigrants/subversives in America call themselves "Americans," these Edomite subversives called themselves Judeans. This is what was meant when Jesus warned that there are those that say they are Jews (Judeans) but do lie. In the original Greek of the New Testament, the word "Jews" doesn't exist, the word is Iudean (Judean). The name "Jews" only began to be used a few hundred years ago.