Untrue - my daughters were born in the states to me (a natural citizen) and my wife (a japanese citizen). She holds dual citizenship until 18, then must decide which as Japan does not recognize dual citizenship in Adults. I have the birth certificates to prove they were born in the states to me and that is all that is required to be a Natural Born Citizen. I have confirmed with legal authority that my interpretations are correct.
What legal authority? There is no case law on the matter. This is not surprising, since the only time the difference, if any, between native born, like your daughters, and natural born (a subset of native born) matters is for eligibility to the office of President.