Three Definitions of United States

Hooven & Allison Co. v. Evatt  324 U.S. 652 (1945)

The term “United States” may be used in any one of several senses. It may be merely the name of a sovereign occupying the position analogous to that of other sovereigns in the family of nations. It may designate the territory over which the sovereignty of the United States extends,  or it may be the collective name of the states which are united by and under the Constitution. [Footnote 6]

[6] See Langdell, “The Status of our New Territories,” 12 Harv.L.Rev. 365, 371; see also Thayer, “Our New Possessions,” 12 Harv.L.Rev. 464; Thayer, “The Insular Tariff Cases in the Supreme Court,” 15 Harv.L.Rev. 164; Littlefield, “The Insular Cases,” 15 Harv.L.Rev. 169, 281

Source: https://supreme.justia.com/cases/federal/us/324/652/case.html