Birth on American soil makes them a citizen....but a "Natural Born Citizen" is something else entirely different. That's why the Constitution grandfathered part of the requirements of Article II; Section I which states
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
I highlighted the part that no longer applies....namely a citizen of the United States.
Natural Born Citizen = Citizen From Birth. There are only two types of citizens, citizens at birth, and citizens who went through a naturalization process. If you are a citizen and not naturalized you are a natural born citizen (Rubio, Cruz, Jindal etc).