The only point in the distinction is in the clause that you leave out: "or a Citizen of the United States, at the time of the Adoption of this Constitution." The reason for that exception is that, by the broadest definition, the oldest natural born citizen of the US was eleven years old in 1787.
If a “citizen at birth” by naturalization statute is (as you contend) a “natural born citizen” at the time of the Adoption of this Constitution what would be the point of Article II’s distinction of “citizen” and “natural born citizen”?