You appear to have never actually read the decision.
Please do so, and pay particular attention to the several times Judge Gray defines natural born citizen.
I have read it several times. In full I might add.
You seem to be missing the SIMPLEST of points. If there was no difference between the two... WHY MAKE THE SPECIFIC DIFFERENTIATION? Hum? If you are talking a single thing, why name it twice and state that one is not the same as the other? Your logic fails. Vattel has it correct, and it is well known (see Benjamin Franklin letters) that the founders had coppies of Vattel.... yea, thats the definition they used.