Naturalization statute may vest citizenship at any point. It may be at birth; or upon the naturalization of the parent(s); or, for those after 18 years of age, upon completion of a process.
A Certificate of Citizenship is issued upon request to a person born outside the United States who derived or acquired U.S. citizenship through a U.S. citizen parent. (documentation requirements must be met)
A Naturalization Certificate is issued to a person who became a U.S. citizen after 18 years of age through the naturalization process.
Citizenship at birth is all that is required to meet the Natural Born Citizen requirement for POTUS.
Twist the rules all you want, but that is the truth of this matter. Twist logic and reason and the language all you want, but the truth remains:
Birthright Citizenship is exactly the same thing as Natural Born Citizenship.