And it proves native-born and natural-born are separate subsets of U.S. citizenship.
No, actually, no, it doesn’t do anything of the sort.
Where does your document state how the two are different from one another?
Does your interpretation appear anywhere? No.
Not really. What it proves, at best, is that whatever bureaucrat was tasked with writing those regulations thought they should say both to cover every eventuality. The government's not monolithic--there's no central Manual of Style with definitions everyone's supposed to use. Do you really think there was some long discussion in the INS offices about Minor v Happersett and Ark and Ankeny and Ramsay vs. Rawls and what those two terms actually meant? Does the document explain the differences anywhere?