"Amendment 23, Section 1 The District constituting the seat of Government of the United States shall appoint in such manner as Congress may direct: A number of electors of President and Vice President equal to the..." They have to use words and phrases like "District constituting the seat of Government of the United States" because in legal documents, you must always use the same words consistently. You don't want people to misunderstand what you mean, in any way. By confusing them with a sudden new word, as in this case. Also, technically Georgetown was part of DC. It was an independent municipality until 1871 when Congress created a new consolidated government for the District of Columbia. But they left that part out.
