From the time of David when Israel was divided into the Northern Kingdom (Israel) which comprised all but Judah and Benjamin, and the Southern Kingdom (Judah), which included only Judah and Benjamin, the name Judah (and particularly "Jew") began to develop into an iconic label for all of Israelite extraction. Much the same way as Kleenex is used generically as a term for all brands of tissue.
1. The division of the land of Israel into tribal territories is set forth in Exodus and Leviticus.
King David was the second ruler of the united Kingdom. The Kingdom split in two, when the Northern Tribes under Ephraim revolted against King Rehobam's taxes.
2. The term "Yehudi" or Jew was used for any subject of the southern kingdom, whether or not he was a member of the tribe of Judah. After fall of the Northern Kingdom and the influx of Northern Israelite refugees, Judah was the only remaining Israelite kingdom. The Northern Israelites who did not flee, where dispersed by the Assyrians and became "lost". (I happen to think that their descendants are Kurds and Pashtuns, but that is another matter.) At any rate, after the return from Babylonian exile, there was another split between Jews and Samaritans. The term "Jew" was used to reffer to any Israelite who followed the Priests centered around the rebuilt Temple on Mt. Moriah in Jerusalem. (Those Israelites and Assyrian settlers who followed the Priests at Mt. Gezerim became known as Samaritans.)
"Actually, the Maronite and Assyrian Christians in Lebanon claim to be Phoenecians"
And I know you do not believe them ;)
The Kingdom was initially divided for a time under David. Avner was instrumental in bringing resolution to the matter. True the division was the house of Saul as opposed to the house of David. So, there was a division.
In any case, please note I said from the "time of David". I intended that era.
The usage of the term "Jew" for all Israel first occurs in the Text during Ahaz's (Hezekiah's father) reign, which was imediately before Hezekiah's reign. If it was in usage at that time, then it is only logical to conclude it predated that time in some effect.
Now, of course, we must date the penning of 2 Kings if we wish for external critical evidence.. but internally, the Text has it just before the time of Hezekiah, and just before the carrying away of Israel to Assyria.
I mostly agree with your point 2. Although I would suggest that Samaritans intermarried with gentiles whereas the "Jews" did not.