So you think Vattel is actually several writers at once? They are obviously referring to many people who wrote on the subject. Vattel is only one of them. Good try.
Obama's parents did NOT have a domicile in the United States in accordance to this definition and therefore fails to be a citizen even under the 14th amendment.
Before you even start arguing domicile, there's the little matter of a mother who was a US citizen.
Vattel is one who the SCOTUS has specifically quoted on multiple occasions. Also from this same ruling, we find Justice Marshall quoting Vattel's definition of natural citizenship with great approval:
Vattel, who, though not very full to this point, is more explicit and more satisfactory on it than any other whose work has fallen into my hands, says"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 indigenes are those born in the country of parents who are citizens. Society not being able to subsist and to perpetuate itself but by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to all their rights."
The court clearly says here that being native = being born in the country to citizen parents. Further it quotes that children naturally follow the condition of their fathers. So much for common law. Under this definition, Obama is a British-Kenyan Subject.
Before you even start arguing domicile, there's the little matter of a mother who was a US citizen.
It is a "little matter" since the Supreme Court has said in at least two different cases that the child naturally follows the condition of the father, one of which I just quoted. Second, why are you bringing up Obama's mama's citizenship if the 14th amendment doesn't look at the citizenship of the parents?? You don't seem to be able to make up your mind as to whether Obama should be a citizen by jus soli or jus sanguinis now.