[Non-Sequitur]

And their is your implicit assumption, for which there is no support in law or the Constitution. The Constitution does about 3 forms of citizenship. Art I, section 8 talks about naturalization, as does the 14th amendment. Art. II Section 1, speaks of Natural Born Citizen, and finally the 14th amendment speaks to "citizenship by birth in the US".

Then what defines the difference between citizen by birth and natural-born citizen?

[El Gato]

Then what defines the difference between citizen by birth and natural-born citizen?

You know that natural born citizen is not defined in the Constitution. Neither is Naturalized citizen, but Congress was given the power to define that. The 14th amendment defines "citizen by birth in the US", but did not change the definition of "Natural Born Citizen". That remains what it was when the base Constitution was passed and ratified in the 1787 to 1788 timeframe.