What law? Seriously? The Naturalization Act of 1795 et seq which naturalize foreign born children of US citizens declaring them in law “citizen”. Bayard’s claim is contrary to the letter of the law.
I’m surprised you cite Bayard. He never claims that naturalized citizens are eligible.
To support your claim that some naturalized citizens are “natural born citizens” and other naturalized citizens are not, you would have been better served by citing the Naturalization Act of 1790.
Only between 1790 and 1795 when the 1790 act was repealed was there such a distinction in law.
It is worth noting that during this period those naturalized as “natural born citizens” were distinguished from the others naturalized as “citizens” by parental US citizenship.
The only time a class of persons was explicitedly declared in law as “natural born citizen” such citizenship was dependant upon parental US citizenship.
The law at the time of Cruz’s birth declared him in law a “citizen”.
Well, I'm glad you understand the law and the Founders' intent better than Chief Justice John Marshall, Justice Joseph Story, Chancellor James Kent, and James Bayard, all of whom were among the most prominent experts in US law and pretty much contemporary with the Founders and Framers - with some of whom they undoubtedly discussed such Constitutional issues.
The law at the time of Cruz’s birth declared him in law a “citizen”.
As did the law at the time that Bayard, Marshall, Story and Kent said that such persons were natural born citizens.