The pamphlet title, or government web page, or opinion of some law professor is .... irrelevant.
What matter is the LAW. The current law is expressed in USC Title 8 Section 1401 - Nationals and Citizens at birth.
Now per the pamphlet you supplied, If one meets the requirements listed, one is a citizen at birth. The record of that birth does not change either the status of that birth nor does it change the individuals citizenship. They were born a citizen and have not needed to be naturalized.
