[Sherman Logan]

the only definition known to mankind at the time, the documented definition by Vattel that excludes people like Obama.

There actually is another definition. It's the definition of "natural-born subject" as provided by Blackstone, the common law authority most familiar to Americans at the time.

Since citizen in a republic is the equivalent of subject in a monarchy, it is not totally unreasonable to assume the Founders would have had this definition in mind.

Under this definition, which has been cited approvingly by the Supreme Court in the few cases that tangentially address the issue, Obama would be NBC (if born in HI).

[PA-RIVER]

David Ramsay, the First American historian who was a president of the Continental congress and friend of Washingtons documents in his book that NBC is the Vattel version, born of citizen parents.

We have the documentation by the founders and Vattel. Natural Born Subject is not mentioned by the founders.
Vattel is the source.