So whenever we read the term Israel in Scripture, how would you know which one the author had in mind?
That understanding comes only through line upon line Bible study, in all of God's Word, even the somewhat boring history parts. (But I like history, so no problem for me.) And that of course is if God gives that understanding. We each must ask Him first, through His Son, and then get down to disciplined Bible study from Genesis all the way to the end of Revelation.
If you study the Old Testament, line upon line, you will discover that God split the original kingdom of Israel into two separate kingdoms (1 Kings 11). He did that because of how Solomon allowed his many pagan wives to bring in their false idol worship into Israel. But He did the split during the days of Rehoboam, the son of Solomon.
"kingdom of Judah" = capital city at Jerusalem. Tribes of Judah and Benjamin at first.
At that split is when God's Word begins to use terms like "Judah", "house of Judah", "Jerusalem", "Judea", "Aholibah", "stick of Judah", put for the southern kingdom.
"kingdom of Israel" = capital city at Samaria in the northern lands. The remaining tribes, ten of them, with one of the tribe of Ephraim God installed as king over them. His name was Jeroboam.
For the northern "kingdom of Israel", God began using terms like "Ephraim", and just "Israel", "Samaria", "Aholah", "house of Israel"; "stick of Joseph".
Because king Jeroboam of the northern "kingdom of Israel" feared many of the ten northern tribes he was over would join with the kingdom of Judah in the south, because Jerusalem is where it was required to worship God at that time, Jeroboam instead setup two golden calf idols in the north, at Bethel and Dan, and told the ten tribes to worship them. He also setup common priests of the people, which made the Levites in the north go south and join with Judah. The southern kingdom thus became mainly the 3 tribes of Judah, Benjamin, and Levi. Small remnants of the northern ten tribes refused Jeroboam's gold calf idols, and they went south also to join with Judah.
The two kingdoms had war against each other.
Then God removed the ten tribe northern kingdom captive to Assyria and the lands of the Medes. They never returned to the holy lands as a people, as God prophesied they would be scattered and lose their heritage as Israel because of their going after Baal and forgetting Him (Book of Hosea). They became lost. Thus the "lost ten tribes" idea. But Bible prophecies puts them mainly in the West, and to there is where The Gospel went to next after Jerusalem rejected it.