That is one definition of citizen.
But the discussion is about eligibility. Not citizenship.
Correct me if I'm wrong, but I believe that "eligibility" to anything is determined by enacted laws and regulations to define your eligibility. Therefore, if you want to talk about "eligibility" to a position that requires one to be a U.S. "citizen", you have to talk about what are the 'laws and regulations' in being a United States "citizen".
The Constitution refers to an individual having to be a "citizen". What kind of citizen? A "natural born Citizen".
You can only become a U.S. "citizen" by either 1 of 2 ways. By "birth" and by "naturalization", which currently provides for the 2 legal citizenship status in this country.
Show me any law or regulation with respect to being a "U.S. Citizen" where you have 3 classes of U.S. Citizenship where you don't fit into the first 2?
Those who want to 'define' a separate distinction for those being BORN HERE as being a "Natural Born Citizen", versus those being BORN HERE but not being a "Natural Born Citizen", are creating a third type of citizenship status that is not defined currently under any legal protection.