You’re correct. The Constitution doesn’t define national born and English Law defines a child born in England by aliens as natural born.
George Mason, delegate from Virginia to the U.S. Constitutional Convention, a "Father of the Bill of Rights" said in the Richmond Convention (debate on the ratification) on Thursday, June 19, 1788 - "The common law of England is not the common law of these States."
The life of George Mason, 1725-1792 (page 431).
The Constitution wasn't intended to be a dictionary. Many things in the Constitution were not defined.
"The English common law provided that an alien naturalized is “to all intents and purposes a natural born subject.” Co. Litt. 129 (quoted and cited in Rhodes, 27 F.Cass. at 790). With such recognition, a naturalized citizen would have been eligible to be President of the new Republic."
If we were going by English law, then Arnold Schwarzenegger would be POTUS eligible.
Chief Justice Marshall, Chief Justice Waite, Justice Story and the House of Representatives that debated the issue of citizenship leading to the 14th Amendment (in particular framer Bingham) on who a "natural born Citizen" is were all in agreement. A natural born Citizen is one born in sovereign territory, to parents (plural) who owe no allegiance to another country.
Barry was born owing allegiance to the crown of her majesty the Queen of England...inherited by birthright from his foreign father.
Barry was a multinational. Citizens can be multinationals (although, the state dept talks of the perils of such), but multinationals and natural born Citizens are mutually exclusive terms.