You assume they referred to Vattel. But we don't know for sure, do we?
Note: I'm not asserting your assumption is in error. You may well be correct. But...we...don't...know...for...sure.
That circumstance leaves it up to Congress to define a citizen. And they have done so: either born in the USA* or born anywhere on the earth with at least one American parent, given certain conditions.
* so-called "birthplace citizenship" for illegals being arguable.
Of course. They can define a citizen however they choose, but only natural law defines a natural born Citizen.
I just assume they meant Natural Born Citizen was something different than Naturalized Citizen. Before Obama most freepers would have agreed with me before they attempted to justify Cruz because Obama got away with it. Just let it go Cruz will never be President, unless he runs as a Democrat.