That's not what the Constitution says. By that standard, John McCain can't be President.
McCain’s status was specifically cleared by Congress when the question first came up. A military installation in the Canal Zone, military parents, is “natural born.” And he has shown his vault version BC.
Childern born outside of the US of personnel in the service of the Federal Government (military, diplomats, etc) ARE considered natural born citizens by law. This has been true since the 18th century.