[MMaschin]

What you seem to be missing, is that 'naturalization' is not a process you go through that ends in a test, and raising your right hand. A 'Naturalized' Citizen is one who requires some positive law for their Citizenship. Reread the SCOTUS opinions in that context, and you will see that, that is consistent with what the court says. What this means, is that someone can be a 'naturalized' Citizen, and the 'naturalization' occurs at birth automatically. So they are both a 'Citizen at birth', and a 'naturalized Citizen'.

A perfect example of this is Ted Cruz. He is both a US citizen at birth, and a naturalized US citizen.

A way to look at this, that might make it easier to see, is to look at it in a negative context. Would the repeal of any law, prior to Cruz's birth, have resulted in Cruz not being a US citizen? The simple answer is yes. This applies to ALL man made laws, even the 14th Amendment.

One additional thing, 'Citizen at birth' does not equate to 'Citizen by birth'. Many people, such as Cruz (and 14th Amendment anchor babies), are 'Citizens AT birth', but the term 'Citizen BY birth' means something completely different, and only 'natural born Citizens' are Citizens BY the fact of their birth alone. So, reread the opinions again, with that understanding, that 'BY birth' does equate to 'natural born Citizen', and 'AT birth' does not.

[Nero Germanicus]

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.