If the principle of "natural born citizen" is based on Vattel's Law of Nations, then this is exactly correct. See below.
If the principle of "natural born citizen" is based on English Common Law, then your children are NOT natural born citizens. Now Jeff is going to come along and say "Yes they are!" and is going to argue that English Common law does make subjects of children born abroad. (Which if true, ought to have made all the Children of British Citizens born in the USA into British Subjects, but paradoxes don't bother Jeff.)
Jeff will be wrong no matter what he says. For one, it isn't English Common law which would be in effect, but a Positive law passed by an act of Parliament which made the Children of Englishmen born abroad into "natural born subjects." (Again, begging the question as to how the Children of British subjects born in America are not British Subjects.)
The point is, British Common law doesn't account for it no matter how you slice it. Vattel's Principles of natural law account for it perfectly.