A natural-born U.S. citizen is defined by Title 8 > Chapter 12 > Subchapter III > Part 1 > § 1401 of the U.S. Code. Your requirement that both parents be citizens is not in there. Neither is it in the Constitution or included in any Supreme Court decision, including the ones you have listed.
The cases I cited above favorably cite Vattel, as his definition informed the Framer’s original intent.
Then you should have no problem providing the quotes from those decisions showing that. Especially Dred Scott, that one has me puzzled.
Dred Scott v. Sandford, 60 U.S. 393 (1857), did not directly deal with Article II “natural born Citizen.” But there are parts of the Dred Scott decision that are relevant to the question of what is a “natural born Citizen.” The case clearly defines “natural born citizen.” While as repugnant as slavery was and still is, no court or amendment has over turned the meaning of “natural born citizen” from Dred Scott. The main point is that in deciding what a “citizen” was in 1817, both the majority and dissent went back to 1787 to examine what the Framers and the people of that time considered a “citizen” to be. The Court said that the Constitution must be understood now as it was understood at the time it was written. The judges did not disagree that one had to look back to the Founding Fathers. What they disagreed on is what the public opinion was at that time as to whether a freed slave was a citizen. In this regard, we know that the case was overruled by the Thirteenth Amendment. As to the “natural born Citizen” clause, the Court said: “The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority, they equally participate in its advantages. The natives, or natural-born citizens, are those born in the country, of parents who are citizens. As society cannot perpetuate itself otherwise than by the children of the citizens, those children naturally follow the condition of their parents, and succeed to all their rights. Again: I say, to be of the country, it is necessary to be born of a person who is a citizen; for if he be born there of a foreigner, it will be only the place of his birth, and not his country. . . .” Dred Scott v. Sandford, 60 U.S. 393, 476-77 (1857).
As can be seen from the quoted language, the Court actually removed from the Vattel’s definition the reference to “fathers” and “father” and replaced it with “parents” and “person,” thus showing that it is not just one parent (the father) that needs to be a citizen, but the “parents,” i.e., both mother and father. Also, both Vattel and the Court state that “if he be born there of a foreigner, it will be only the place of his birth, and not his country.” The controlling language is “a foreigner.” In the English language, the letter “a” is an indefinite article meaning one. Hence, the use of the word “a” shows that only one is required. We know that a child has both a mother and father and the “a” would necessarily refer to either the mother or father. Surely, if the child were born of one parent who was not a citizen, he would be “born there of a foreigner,” who would be either his foreign mother or father. As can be seen, it is our United States Supreme Court that has made this reading and interpretation of Vattel. This understanding of Vattel is not only correct but also binding upon us.
Source: http://puzo1.blogspot.com/2009/09/natural-born-citizen-clause-requires.html