For example, Interpretation 301.1 United States citizenship doesn't make this same distinction.
http://www.uscis.gov/ilink/docView/SLB/HTML/SLB/0-0-0-1/0-0-0-45104/0-0-0-45113.html#0-0-0-22819
(a) Birth in the United States . (1) Statutory development . Prior to 1866, absent any statutory or constitutional provision, it was generally held, under the common-law principle of jus soli (the law of the place), that a person born in the United States acquired citizenship at birth; this principle was incorporated in the Civil Rights Act of April 9, 1866, and, two years later, found expression in the Fourteenth Amendment to the United States Constitution, which provides that all persons born in the United States, and subject to its jurisdiction, are citizens of the United States.
The above constitutional provision has remained in effect ever since, and is restated in this section. 1/
(2) “United States” defined . Prior to January 13, 1941, the term “United States” included the continental mainland, Hawaii after August 11, 1898, 2/ Alaska, upon its formal incorporation into the Union on March 30, 1867, 3/ but not Puerto Rico. 4/ The Philippine Islands have never been deemed to be part of the United States within the purview of the 14th Amendment. 4a/
The territorial limits of the United States were extended on January 13, 1941, to include Puerto Rico and the Virgin Islands, 5/ and the current statute has added Guam to this definitive grouping 6/
In addition to the land areas mentioned above, ports, harbors, bays, enclosed sea areas, and a three-mile marginal belt, along the coasts thereof, form a part of the territorial limits of the United States. 7/
Notwithstanding the position taken in the second paragraph under INTERP 316.1(b)(2)(i) and in INTERP 329.1(c)(3)(ii) , Midway Islands is not and never has been considered a part of the “United States” in the sense that United States citizenship is or was acquired at birth therein under the Fourteenth Amendment to the United States Constitution, or any of the various statutory provisions which have conferred citizenship upon such basis.
(3) Effect of parents’ status . Alienage of a child's parents does not preclude his acquisition of citizenship jus soli nor did their racial ineligibility for naturalization under former laws have such result. 8/
Additionally, acquisition of citizenship is not affected by the fact that the alien parents are only temporarily in the United States at the time of the child's birth.
What you found here is an interpretation on the presumed citizenship of foundlings. It would make sense they would only be presumed to be native-born. Without knowing whether the parents are citizens, they can’t be presumed to be natural-born. Your other findings generally don’t need to make a distinction between natural-born and native-born, since as cited, the former would be a subset of the latter. What Leo found requires a different distinction because it involves restoring persons to their previous status, which obviously could be THREE different things: naturalized, native-born or natural-born.