When the area you were born or live in changes governments, does that make you an immigrant?

It's in the Constitution: "No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President". - Article II, Section 1, Paragraph 5

As you can see, this covers the corner case you've identified.

Interestingly, the term citizen wasn't defined until the 14th amendment: "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside"