That may be the way you want it to be, and that certainly is the way I want it to be, but that's not the way it currently is.
The Supreme Court in United States v. Wong Kim Ark answered the question that anyone born in the United States is a United States Citizen.
Furthermore, the Fourteenth Amendment only recognizes two types of citizenship; natural born and naturalized. There is no third category that you are trying to create.
So until we have a Constitutional Amendment clarifying that children born in the US to non-US citizen parents do not receive US citizenship, or that children born of foreign parents are considered "naturalized" citizens, Haley, Jindal, and Rubio are legally natural born citizens.
Don't like it, but that's the way it is.
Here’s the text of Section ! of the 14th Amendment:
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
You’ll notice there is no reference or use to the term “natural born citizen”, only “citizens”.
The 14th Amendment did not overturn, nor can it overturn the natural born citizen requirement for President in the original text of the Constitution.
Natural born citizen is a defined term, both at the time of the creation of the US Constitution, and in Minor v. Haperstett 1875.
Natural born citizen is one born in country of two CITIZEN parents, or out of country of the two
CITIZEN parents are in the servidce of the government.
You can be a citizen in one of three ways: born a citizen within the US, naturalized but born on another country, and natural born, born here of citizen parents.
The law does not say that natural born equals born citizen in any case.
If your parent(s) are not citizens of the US, and you are born here, you are a citizen, but not a natural born citizen, thus ineligible to be President or Vice President.