No, just saying that the faith they follow is the faith of Abraham, Isaac, Jacob and Jacob’s 12 sons. Anyone can follow that faith as did the Samaritans of the time of Jesus (who were looked down upon by the Jews). But to be a Jew, you must be from the line of Judah. Others are followers of the Jewish faith.
Just like I could move to China and learn the language and culture but I would never be Chinese.
Think of the Jewish people as a family. A convert is adopted into the family with all of the rights and responsibilities.
A convert is a true Jew.
Just ask Ruth...
You are wrong. The term "Jew" has been universally applied to all Israel since the days of Ezra. Ask any Cohen or Levi today (both are from the Tribe of Levi, not Judah). They'll identify as Jews, and every Jew on the planet would recognize them as such. Heck, consult your Christian Bible. Paul was from the Tribe of Benjamin. But he clearly identifies as a Jew.
“But to be a Jew, you must be from the line of Judah. Others are followers of the Jewish faith.”
So me, a lowly Cohen of the Tribe of Levi, is not Jewish?
Sorry, you may have a point in the original technical sense of the word, but for 2500 years or so, that’s a distinction without a difference.
No, one does not need to trace his ancestry back to Biblical times in order to be Jewish. Many of today's Jews, whether they know it or not, are descended from converts to Judaism somewhere along the way.
And one does not necessarily have to be a "follower of the Jewish faith" to be Jewish, since Jews are both a national/ethnic group and a religious group, as distinguished from, e.g., Christians and Muslims. However, if one born to a Jewish mother formally converts into another religious tradition, he would no longer be considered Jewish according to Jewish law.