That act was repealed and its replacement removed the mention of natural born citizen.
Even in the language of that repealed act, it stated that children born of citizens abroad “shall be considered as natural born citizens”, clearly indicating that they were not.
It is my understanding that that statute was not actually "repealed" but simply superseded by a new statute with a new definition. And that raises the question whether a natural born citizenship can be fixed by act of Congress?
In any event the first statute is evidence of the understanding of the first Congress about the meaning of the phrase "natural born" which favors Ted Cruz. There are cases that seem to go the other way as does the subsequent statute you cite.
Please forgive if I have any of these facts wrong, all of this is from memory of some time ago.
