A natural-born citizen is a citizen from time of birth. Those are synonymous terms. Any definition of a citizen from time of birth is therefore describing a ‘natural-born citizen’.
The only other type of citizen is a naturalized citizen.
The Presidential eligibility reqt is to exclude naturalized citizens.
This is wrong. There are several types of statutory citizens at birth that clearly aren't natural born citizens. Natural born citizenship is such that you are born with direct ties to the country where you were born. The presidential eligibility requirement is to exclude all persons born with ties to foreign countries.