“Why would he stipulate natural born over just citizen if they were the same?”
Because they are not all the same. There are citizens born and citizens made. A naturalized citizen cannot be President.
U.S Citizenship Law, 8 USC Section 1401 naturalizes persons as U.S. Citizens at birth if they meet on of the circumstances listed therein. Naturalization as a Citizen can be granted by the United States under the powers of Congress per the U.S. Constitution, either at birth or later in their life under the relevant legal process. I suggest you read the Constitution as to what type of Citizenship the Congress can grant under laws or amendments it would pass. Congress has only naturalization powers to make Citizens by man-made Positive Law. And no where in that man-made naturalization at birth law is the word “natural” used nor is the term “natural born Citizen” used. A naturalization law is man-made law and said man-made Positive Laws can only create or grant being a Citizen by naturalization. Only Natural Law and the Laws of Nature can create a “natural born Citizen”.
And Vattel’s treatise on Principles of Natural Law which was widely read and used by the founders and framers and early Justices of the Supreme Court, clearly and succinctly defined who is a “natural born Citizen”, i.e., one born in the country to parents [plural] who are Citizens of the country. See: https://lonang.com/library/reference/vattel-law-of-nations/vatt-119/