Within the boundaries of the United States there were at various times "organized incorporated territories", that were not a part of any of the states and whose governments were organized by acts of Congress rather than by a state constitution drafted by statesmen within the state and enacted by the voters of the state, and that, unlike the states, had no voting representatives or senators in Congress. (Somewhat like the three territories of northern Canada today, I think?) Is the District of Columbia simply an instance of that phenomenon, or is there some essential difference?
Art. I Sec. 8 Cl. 17 states that "The Congress shall have Power"
To exercise exclusive Legislation in all Cases whatsoever, over such District (not exceeding ten Miles square) as may, by Cession of particular States, and the Acceptance of Congress, become the Seat of Government of the United States, and to exercise like Authority over all Places purchased by the Consent of the Legislature of the State in which the Same shall be, for the Erection of Forts, Magazines, Arsenals, dock-Yards, and other needful Buildings
which then happened in 1 Stat. 130 (1790). It is there referred to as "a district of territory" but more often simply as a "district".
Unlike territories, its existence for its actual purpose as seat of government is specifically enabled by the Constitution. Then via 2 Stat 103 (1801), DC was politically brought within the control of Congress, so that residents were no longer residents of Maryland or Virginia. In this act it is consistently termed a "district".