“State Department says born of US parents on foreign soil is a US citizen.”
A U.S. Citizen can’t become president anymore according Article 2 Section 1 but a natural born Citizen born within the jurisdiction of the U.S. can. Only at the time of the adoption of the Constitution could a Citizen become a president.
Article 2 Section 1
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; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
You added a requirement I don't see in the Constitution.