[MamaTexan]

Please show me where in the Constitution that distinction is set down.

When the Constitution was written, it was based on over 200 years of English Law. One of the rules operating under that law was the Rule of Exclusion.

§ 207. XIII. Another rule of interpretation deserves consideration in regard to the constitution. There are certain maxims, which have found their way, not only into judicial discussions, but into the business of common life, as founded in common sense, and common convenience. Thus, it is often said, that in an instrument a specification of particulars is an exclusion of generals; or the expression of one thing is the exclusion of another. Lord Bacon's remark, "that, as exception strengthens the force of a law in cases not excepted, so enumeration weakens it in cases not enumerated," has been perpetually referred to, as a fine illustration.
Justice Joseph Story on Rules of Constitutional Interpretation

It is a general principle of statutory construction that the mention of one thing implies the exclusion Of another. (For an exhaustive annotation on this "rule of exclusion," see: 73 Am. Jur. 2d Statutes §212 and the citations collected thereunder.) As exceptions in a statute strengthen the force of law in cases not excepted, so enumerations weaken it in cases not enumerated.
John C. Danforth, Missouri Attorney General, April 21, 1975

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Enumeration combined with the Rule of Exclusion is what makes the Constitution a limiting document!

Since being natural born is enumerated as a requirement for President, any other type of citizen (such as naturalized or native born) is automatically EXCLUDED.

[sometime lurker]

You cite Justice Story. Perhaps you missed his opinion in Inglis v. Trustees of the Sailor's Snug Harbour:

he was born an American citizen, whether his parents were at the time of his birth British subjects or American citizens. Nothing is better settled at the common law than the doctrine that the children even of aliens born in a country, while the parents are resident there under the protection of the government, and owing a temporary allegiance thereto, are subjects by birth.

The Constitution only mentions two types of citizens - natural born and naturalized. There is no third category. Many sources use "natural born" and "native born" interchangeably.