[nathanbedford]

It is my understanding that that statute was not actually "repealed" but simply superseded by a new statute with a new definition.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.

[Lurkinanloomin]

No, acts of Congress cannot change the Constitution. That is why the first act was repealed and its replacement removed the natural born citizen language. They realized their mistake.
Children of foreign nationals are not natural born citizens even if born here.
Children of foreign nationals born abroad are not even citizens unless the US citizen parent applies for recognition.
When did Cruz or his parents file for a CRBA?