There are only two types of citizen: natural born and naturalized. Arthour was a citizen by birth not naturalization.
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.
My last post to you should have had a paragraph mark before the second heading “Naturalized citizenship.” That would help with clarity.
Also, I’ve resorted to shorthand at times in my posts on this. I want it to be clear that “by descent” still leaves open the question of whether, at the applicable time, “natural born” status required descent from the father, or from the father or the mother, or from both.