I take it that ‘CitizenS’ refers to/means that both parents have to be ‘citizens’. Do you have a reference that Bayard did not mean such? All the judges and experts surely could/would have made any/some differentiation in writings you reference.
Yes. It's in the statement he made.
If you're "born a citizen," then as far as the American legal system and the experts of the early United States are concerned, you're a natural born citizen.
That's all it means.
So if you were born a citizen because you had one citizen parent, like Cruz, you're good to go.
The Founding Fathers wanted to exclude anyone from the Presidency who was not born a United States citizen.