Those statements are not confirmed under law. As I said Black’s does not define “native born” as being anything different than “natural born”.
Congress is given the authority under the Constitution to create the mechanism to become a naturalized citizen.
But they were not given the power to define "natural born citizen". The statutes do not mention "natural born". The only that did was the 1790 immigration law, which was repealed and replaced by the 1795 law, which dropped the words "natural born" when speaking of persons born outside the US of citizen parents. Why did they change it? Well, although I don't know, and probably no one else does either, I speculate that they figured out they could not redefine a Constitutional term under their power to define an uniform rule of naturalization. Which by the way, was the title of both the 1790 and 1795 acts, "An Act to Define an Uniform Rule of Naturalization".