for a long time, texas was a part of mexico. the disputed area represented the land that was technically part of mexico but wanted to be owned by the US. due to this, congress declared war on mexico in 1846 in hopes of gaining this territory as its own.
i would label america saying that they wanted to gain control over that part of mexico.
i would label mexico as having control over area that is most commonly known as modern day texas
and, i would label the disputed area as the territory fought over between the us and mexico during the mexico-american war
hope this helps!!