Alright, let's go with your terminology then:
There are two types of citizenship status:
(1) Natural born citizenship. This status is attained upon birth by operation of nature (that is, by descent). No law has ever been required to deem the child of two Americans an American citizen, regardless of place of birth. (2) Naturalized citizenship. There are two types (subsets) of naturalized citizenship:
(a) Automatic naturalization at birth. This status is attained upon birth by operation of law. The children of aliens born within a country / kingdom were deemed citizens of that place solely because the sovereign chose to accord them citizenship. Sovereigns could have made a different choice, therefore proving that this status comes only through law, not through nature.
(b) Voluntary naturalization some time after birth. This status is attained later in life by operation of law and volition. As such, it also is not a status that occurs by nature.
While "natural" citizenship status and "automatic naturalization" status are both attained upon birth, they are not the same and should not be confused.
That said, there is only one difference between the "natural" status and the "automatic naturalization" status: that is, that only the former makes one eligible to serve as President.
The framers' goal was to limit the presidency, and particularly the office of Commander in Chief, essentially to those who were at least second-generation Americans. A child who is automatically naturalized at birth---i.e., who is an American because American law says he is, not because his parents were Americans---is a first-generation American. That was too close to the old country for the framers.
Naturalization at birth? That's the first I've heard of that notion. Is there law or case law on it?
I had thought that the word naturalization was coined to denote the process of an existing non-citizen becoming a citizen, as opposed to being born a citizen.
Is Governor Jindal a natural born citizen? Or is he naturalized? Can he be president? His mother arrived in this country four months pregnant, and his Indian grad student parents (I'm 99% certain) were not yet naturalized when he was born in Louisiana. Is he your category (1) or (1a)? Is he natural born or naturalized? Does his parents having subsequently become naturalized citizens retroactively make him natural born?