Do foreign born children of American citizens go through a "process"?
No. But they are still naturalized. They are made citizens by one of the naturalization acts of Congress, such as the naturalization act of 1952, which is how Barack may have gotten whatever sort of citizenship he actually has, if any.
The 14th amendment is a naturalization law. If you are a citizen only because of the 14th amendment, you are a naturalized citizen, even though no one treats you as if you are a naturalized citizen.
But you can go look at the debates in Congress on the 14th amendment, and they will tell you, that the 14th amendment is naturalization.
It is the making of people into citizens by a law passed by congress.
A "natural born citizen" is someone who, by operation of law -- i.e., whatever law was in place at the time of the person's birth -- was a U.S. citizen at birth. That's all "natural born citizen" means, and that's all the term has ever meant.
That's incorrect. That's just the oft repeated modern claims by people who wanted Barack Obama to be legal.
If you look in history, you find out this isn't correct. Here is an example.
Law book from 1817 Pennsylvania.