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For your consideration:
From the U.S. government’s brief in U.S. v. Wong Kim Ark (1898):
“Are Chinese children born in this country to share with the descendants of the patriots of the American Revolution the exalted qualification of being eligible to the Presidency of the nation, conferred by the Constitution in recognition of the importance and dignity of citizenship by birth?” If so, then verily there has been a most degenerative departure from the patriotic ideals of our forefathers; and surely in that case American citizenship is not worth having.”

http://librarysource.uchastings.edu/library/research/special-collections/wong-kim-ark/AppellantsBrief.pdf
The Government (Appellant) Brief: US v Wong Kim Ark (page 18 of 20 of the pdf) Page 34 of the original

From the Supreme Court’s majority decision in U.S. v. Womg Kim Ark:
[An alien parent’s] “allegiance to the United States is direct and immediate, and, although but local and temporary, continuing only so long as he remains within our territory, is yet, in the words of Lord Coke in Calvin’s Case, 7 Coke, 6a, ’strong enough to make a natural subject, for, if he hath issue here, that issue is a natural-born subject’
‘Subject’ and ‘citizen’ are, in a degree, convertible terms as applied to natives; and though the term ‘citizen’ seems to be appropriate to republican freemen, yet we are, equally with the inhabitants of all other countries, ’subjects,’ for we are equally bound by allegiance and subjection to the government and law of the land.’

and

…every child born in England of alien parents was a natural-born subject, unless the child of an ambassador or other diplomatic agent of a foreign state, or of an alien enemy in hostile occupation of the place where the child was born.

The same rule was in force in all the English colonies upon this continent down to the time of the Declaration of Independence, and in the United States afterwards, and continued to prevail under the constitution as originally established.”

Kwock Jan Fat v. White, 253 U.S. 454 (1920)

It is not disputed that if petitioner is the son of Kwock Tuck Lee and his wife, Tom Ying Shee, he was born to them when they were permanently domiciled in the United States, is a citizen thereof, and is entitled to admission to the country. United States v. Wong Kim Ark, 169 U. S. 649. But, while it is conceded that he is certainly the same person who, upon full investigation, was found, in March, 1915, by the then Commissioner of Immigration, to be a natural born American citizen…

Perkins v. Elg, 307 U.S. 325 (1939)

“And the mere fact that the plaintiff [Elg] 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….

The court below, properly recognizing the existence of an actual controversy with the defendants [page 350] ….. declared Miss Elg “to be a natural born citizen of the United States,”

Marie Elizabeth Elg was born in the Brooklyn section of New York City in 1907 to two Swedish parents who had arrived in the United States some time prior to 1906; her father was naturalized in 1906. In 1911, her mother took the four-year-old to Sweden; her father went to Sweden in 1922, and in 1934 made a statement before an American consul in Sweden that he had “voluntarily expatriated himself for the reason that he did not desire to retain the status of an American citizen and wished to preserve his allegiance to Sweden.”
In 1929, within eight months of attaining the age of majority, Marie Elg obtained an American passport through the American consul in Sweden, and returned to the United States. In 1935 she was notified by the U.S. Department of Labor that she was an illegal alien and was threatened with deportation.
Elg sued to establish that she was a citizen of the United States and not subject to deportation. Frances Perkins was listed as the nominal plaintiff in the case, being the Secretary of Labor during the administration of Franklin D. Roosevelt, when the case was appealed to the Supreme Court.

Here the US Supreme Court equates citizenship by birth with natural born citizenship:
Elk v Wilkins, 112 U. S. 94 (1884):
“The distinction between citizenship by birth and citizenship by naturalization is clearly marked in the provisions of the constitution, by which ‘no person, except a natural-born citizen, or a citizen of the United States at the time of the adoption of this constitution, shall be eligible to the office of president;’ and ‘the congress shall have power to establish an uniform rule of naturalization.’Const. art. 2, § 1; art. 1, § 8.”

I will respond to this as time permits.

Firstly, Kwock Jan Fat v. White, 253 U.S. 454 (1920)

This was a ninth circuit case reversed by the Supreme Court.

From your post:

Now let's take a look at the full text without the chopped sentence:

But, while it is conceded that he is certainly the same person who, upon full investigation, was found, in March, 1915, by the then Commissioner of Immigration, to be a natural born American citizen, the claim is that that Commissioner was deceived, and that petitioner is really Lew Suey Chong, who was admitted to this country in 1909 as a son of a Chinese merchant, Lew Wing Tong, of Oakland, California.

Context can be everything.

Kwock Jan Fat v. White - 253 U.S. 454 (1920)