The idea that ‘citizen at birth’ equates to ‘natural born citizen’ ignores the word ‘natural’.
If the intention was otherwise, they would have simply said a ‘born citizen’, or a ‘citizen at birth’ or ‘born a citizen’.
So it is clear they intended something else. So - what does the word ‘natural’ mean in the context of ‘natural born citizen’? There are two types of law. There is ‘positive law’ - this is man-made law, such as the Constitution, laws from Congress, state law, local ordinances, and so on. And then there is ‘natural law’ - this is the law of nature, or the divine."
Indeed. Simple logic dictates this.
And, speaking of natural law: Our Declaring Independence, and the founding document itself is based upon natural law principles.