That makes sense. To your knowledge was it ever used as that after the split or are all those references after Solomon to the 10 tribes only? If they weren't I have certainly been off base with some things!
......Ping
I think the rule of thumb is: Whenever the two houses are mentioned together (Israel and Judah) in the same thought or passage....they are to be considered distinct. Whenever only one is mentioned they are to be considered only one. The problem some folks have though.....is recognizing that Judah is still part of Israel and when Israel is mentioned by itself you should always consider the fact that this includes Judah.
[Jeremiah 3:6-18] is a good example of the separate distinctions of the two houses.....whereas [Revelation 7:4-8] is a good example of The Holy Spirit wanting us to include Judah in the House of Israel (Tribes of Israel).
[Matthew 15:24] But he answered and said, I am not sent but unto the lost sheep of the house of Israel. This is one of the few times in the New Testament that the "House of Israel" is referred to. Most of the time they are called "Children of Israel" in the New....except for a few passages in Hebrews 8. As you can see....Our Lord is including Judah with this statement.
Sometimes scriptures will speak of "Ephraim" in the same vein as Israel (Ten Tribes). Ephraim was the leading tribe of the north. Many passages in Hosea show this distinction.