Natural born citizenship is MORE than the citizenship at birth you describe, it is citizenship by NATURE and not by mere statute. For you to ignore this is to be in denial.
And citizenship by birth is in fact not about a statute, so your point doesn't make sense.
A person born in the United States is a natural born citizen. It has been the case since the founding of this country, and before under the English system. There is not a single legal authority, no law, no court ruling, nothing, contradicting this. It is just something birthers have latched onto because it fits their desire.