Contrary to what some allege, Perkins v. Elg very clearly states that native and natural born are the same:
See http://openjurist.org/307/us/325/perkins-v-elg-elg
(Excerpt from SCOTUS decision):
‘Young Steinkauler is a native-born American citizen. There is no law of the United States under which his father or any other person can deprive him of his birthright. He can return to America at the age of twenty-one, and in due time, if the people elect, he can become President of the United States; ...”
and
(Excerpt from SCOTUS decision):
“First.—On her birth in New York, the plaintiff became a citizen of the United States. Civil Rights Act of 1866, 14 Stat. 27; Fourteenth Amendment, Section 1, U.S.C.A.Const.; United States v. Wong Kim Ark, 169 U.S. 649, 18 S.Ct. 456, 42 L.Ed. 890. In a comprehensive review of the principles and authorities governing the decision in that case—that a child born here of alien parentage becomes a citizen of the United States—the Court adverted to the ‘inherent right of every independent nation to determine for itself, and according to its own constitution and laws, what classes of persons shall be entitled to its citizenship.’ United States v. Wong Kim Ark, supra, 169 U.S. page 668, 18 S.Ct. page 164, 42 L.Ed. 890. As municipal law determines how citizenship may be acquired, it follows that persons may have a dual nationality.1 And the mere fact that the plaintiff may have acquired Swedish citizenship by virtue of the operation of Swedish law, on the resumption of that citizenship by her parents, does not compel the conclusion that she has lost her own citizenship acquired under our law. As at birth she became a citizen of the United States, at citizenship must be deemed to continue unless she has been deprived of it through the operation of a treaty or congressional enactment or by her voluntary action in conformity with applicable legal principles.”
Read the whole decision at the URL above.
You're citing case law to people who view this not as a matter of fact, but as an article of faith. It's futile. They will believe what they wish to believe, and any volume of unimpeachable and incontrovertible fact will be dismissed as "trolling" or subterfuge.
When people would rather base their positions on hearsay, supposition and urban legend, facts become irrelevant, as does anything that impedes their political narrative they cling so tightly to.
‘Young Steinkauler is a native-born American citizen. There is no law of the United States under which his father or any other person can deprive him of his birthright. He can return to America at the age of twenty-one, and in due time, if the people elect, he can become President of the United States; ...”
The Supreme Court qualified young Steinkauler's status when they said his father became a US citizen in 1854 where the young Steinkauler was born in St Louis a year later. They need not mention his mother because the mother was either a US citizen already or the mother received her US citizenship by virtue of marriage to her husband. In the lexicon of the Supreme Court to describe Steinkauler as a 'native born' would be appropriate and without confusion after they gave a background to his father's citizenship status because all 'natural born' citizens are 'native born', but NOT all native born are natural born citizens.
Excerpt from SCOTUS decision):
‘Young Steinkauler is a native-born American citizen. There is no law of the United States under which his father or any other person can deprive him of his birthright. He can return to America at the age of twenty-one, and in due time, if the people elect, he can become President of the United States; ...”
If you read the case you will find that the father was naturalized and therefore a citizen before the birth of, “...Young Steinkauler,” making him a natural born citizen. I suggest you re-read your examples with the understanding of what constitutes a natural born citizen, which is, a person born of two citizen parents in the USA, or outside, while serving in the military or consular service.