You’ve done an excellent job of explaining that the word “naturalization” has multiple meanings. All citizens are naturalized but non-citizen, foreign born immigrants with no previous ties to the US can go through a statutory process of becoming naturalized US citizens.
As the Supreme Court has ruled in Schneider v. Rusk (1964),
“We start from the premise that the rights of citizenship of the native born and of the naturalized person are of the same dignity and are coextensive. The only difference drawn by the Constitution is that only the ‘natural born’ citizen is eligible to be President. Art. II, s 1.”
Yes the Court used the colloquial term “native born” to be synonymous with “natural born.”
It’s really quite simple, Citizens of the United States at Birth can be President and naturalized citizens cannot be president. There is no distinction between a Citizen of the United States at Birth and a Natural Born Citizen that has ever been found in court rulings or statute.
The Supreme Court has not made a distinction based on choice of preposition, “at” or “by” birth.
For example: In 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.”
SCOTUS went on to say: “This section contemplates two sources of citizenship, and two sources only: birth and naturalization. The persons declared to be citizens are “all persons born or naturalized in the United States, and subject to the jurisdiction thereof.”
There is no distinction in case law, administrative law or statutory law between a Citizen of the United States AT Birth and a Citizen of the United States BY Birth. Only the preposition “at” is used in the US Code.
And what did the court mean by usage of the word "Native" in say, 1875?
At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners.
Funny, their definition of the word "native" corresponds to OUR definition of the word "natural born" but not yours. How did the Justice Waite every become Chief Justice without understanding the meaning of this term "native"?
It's a mystery! Obviously the 1875 court is wrong, not you.