Why do you suppose the Constitution draws a distinction between the President, who must be a "natural born citizen" and the Senate who must be a "citizen"?
There is this:
During the process of developing a new U.S. Constitution Alexander Hamilton submitted a suggested draft for a Constitution on June 18, 1787. He also submitted to the framers a proposal for the qualification requirements in Article II as to the necessary Citizenship status for the office of President and Commander in Chief of the Military.
Alexander Hamilton’s suggested presidential eligibility clause:
"No person shall be eligible to the office of President of the United States unless he be now a Citizen of one of the States, or hereafter be born a Citizen of the United States."
John Jay wrote in a letter to George Washington dated 25 Jul 1787:
"Permit me to hint, whether it would be wise and seasonable to provide a strong check to the admission of Foreigners into the administration of our national Government; and to declare expressly that the Commander in Chief of the American army shall not be given to nor devolve on, any but a natural born Citizen. "
Article II, Section 1, Clause 5 of U.S. Constitution as adopted 17 Sep 1787:
"No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States."
Why do the authors use the term "natural born citizen" distinctly from "citizen"? They seem to understand it among themselves, where did they get it and what did it mean to them?
What did this cite mean?:
"The Constitution does not, in words, say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that. At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners.
Chief Justice Morrison Waite, from Minor v. Happersett, cited by Justice Horace Gray as the meaning of natural born citizenship in United States v Wong Kim Ark
Frequently however, the law and "logic" are two disparate characters in the same play.
You make the same argument many people make here. Then, you go on to quote the Constitution or Vattell or someone else. As a legal argument, it's wholly unpersuasive. If you can point to a precedential Court ruling on the central legal argument - that anchor babies aren't natural-born citizens, then that would be persuasive. You can't.
Until the Supreme Court says that "anchor babies" aren't natural-born citizens, then as a practical legal matter, they are. Moreover, Barack Obama is not an "anchor baby". He is quite clearly a man born on US soil to at least on US citizen parent.