As I understand it, the distinction between the Indian Territory and as US Territory has to do with political organization. The US certainly claimed all the land from coast to coast as territory of the US (manifest destiny) and the Indian Territory was acquired as part of the Louisiana Purchase. However, the US did not exert political control over the Indian Territory and US citizens were not supposed to live within the Indian Territory without special permission. There was no territorial governor over the Indian Territory.
Originally, the term Indian Territory was used rather loosely to describe all the western lands occupied by the Indians. As "US Territories" were created -- Kansas Territory, Nebraska Territory, etc. -- the "Indian Territory" became smaller and eventually became what is now Oklahoma.
The US did not exert political jurisdiction over the Indian Territory but they did impose military jurisdiction and became more and more intrusive into tribal politics. After the war, the new treaties with the US resulted, as did nearly all treaties, in land concessions which led to creation and growth of Oklahoma Territory which contained Indian "reservations".
The so-called Five Civilized Tribes were never on "reservations", they own their land and now share joint jurisdiction with the State of Oklahoma. The tribes that were moved to Oklahoma after the war were placed on reservations and declared "wards of the US Government". The US Supreme Court ruled (Cherokee Nation vs the State of Georgia) that the Five Civilized Tribes are "domestic dependent nations", not "wards of the US Government".
I assume the Unassigned Lands you are referring to are what is now the Oklahoma panhandle. These were not US Territory either -- they were Indian Territory, just not assigned to a particular tribe. Originally, the Cherokee Treaty provided for an unbounded outlet to the west for hunting which theoretically extended to the west coast so the Unassigned Lands were originally Cherokee. Later the western boundary was established of what was/is commonly (but incorrectly) called the Cherokee Strip -- the correct name is Cherokee Outlet. (The Cherokee Strip was a two mile strip along the Kansas border where Congress accidently on purpose designated the southern boundary of Kansas Territory to be two miles south of the Cherokee Nation's northern boundary.)