The original point was that it was de Vattel's 'Law of Nations' published in 1792 since you were making the ten years [1797] after the 1787 Constitutional Convention silliness and then it was clarified for you in subsequent posts.
They "rejected de Vattel's idea"? Hahaa... of course the Law of Nations was not. Benjamin Franklin didn't reject it as a signatory to the US Constitution.
To refresh your memory again, here is Benjamin Franklin's own words to Charles Dumas.
"I am much obliged by the kink present you have made us of your edition of Vattel. It came to us in good season, when the circumstances of a rising State make it necessary frequently to consult the law of nations. Accordingly, that copy which I kept (after depositing one in our own public library here, and sending the other to the College of Massachusetts Bay, as you directed) has been continually in the hands of the members of our Congress now sitting, who are much pleased with your notes and preface, and have entertained a high and just esteem for their author...."
Franklin brought back 3 copies of the Law of Nations from France to the new continent. It was "continually in the hands of members" of Congress? And those same Founding Fathers praised de Vattel speaking about him in "high and just esteemed"? Oh yeah, it was 'rejected' in your words - LoL!
As to your quotation from Cheif Justice Fuller, I suppose that you did not notice he was writing in the dissent? In other words, this is the opinion that lost the decision 6-2.
So, rather than showing that "the Supreme Court of the United States believed that de Vattel's definition is the meaning and intent behind Article 2, Section 1," it is instead a demonstration of what the Supreme Court, represented by the majority and the deciding law did not believe.
You didn't notice - again. Take a look at the post again to you that I said that these words were not lost of Justice Grey who wrote the majority opinion for the court. As Grey did not say in the majority court's judgment that Wong Ark was a natural born citizen which is the point. Without the minority opinion of the court who reminded and let facts be known to the majority, the majority could have made an even larger egregious error in their opinion.
I hafta tell you... it is a very weird debate tactic to draw so much attention on the fact that your preferred definition didn't just lose, but lost explicitly after having been considered by the court.
It's well explained in the above post to you, and the point being made did not lose in court - Ark and Obama are not natural born citizens. Furthermore, what's weird is your misdirection,...then again on second thought, I expect it.