Consider this: on day 1 the Constitution spoke only of "people", "citizens" and "natural born citizens". It further recognized and included language that made provision for the fact it would be several decades before it would be possible for anyone to be a "natural born citizen".
With regard to your contention that since birth certificates do not provide for a "natural born" option, there must not be such a classification, bear in mind certificates have generally requested the birthplace of both parents.
Current immigration policies even today provide for some to be "natural born citizens" (google it yourself, don't take my word for it.)
The Supreme Court, since the ratification of the 14th Amendment in 1868 (“all persons born or naturalized...”) has come down on the side of only two forms of citizenship. For example: “United States v. Wong Kim Ark (1898)
[An alien parent’s] allegiance to the United States is direct and immediate, and, although but local and temporary, continuing only so long as he remains within our territory, is yet, in the words of Lord Coke in Calvin’s Case, 7 Coke, 6a, ’strong enough to make a natural subject, for, if he hath issue here, that issue is a natural-born subject’
‘Subject’ and ‘citizen’ are, in a degree, convertible terms as applied to natives; and though the term ‘citizen’ seems to be appropriate to republican freemen, yet we are, equally with the inhabitants of all other countries, ’subjects,’ for we are equally bound by allegiance and subjection to the government and law of the land.’
Every child born in England of alien parents was a natural-born subject, unless the child of an ambassador or other diplomatic agent of a foreign state, or of an alien enemy in hostile occupation of the place where the child was born.
The same rule was in force in all the English colonies upon this continent down to the time of the Declaration of Independence, and in the United States afterwards, and continued to prevail under the constitution as originally established.”
Yes, these are different classifications. "people" is everyone, "citizens" are all citizens, "natural born citizens" are that sub-class of citizens that are citizens by birth, not naturalization.
"It further recognized and included language that made provision for the fact it would be several decades before it would be possible for anyone to be a "natural born citizen".
Yes, because obviously on July 4, 1776 nobody had been born in the United States. Their parents had nothing to do with it.
"...bear in mind certificates have generally requested the birthplace of both parents."
Because birth certificates are identity documents and establish familial relationships. They also include birthplace, which is sufficient for citizenship questions.
"Current immigration policies even today provide for some to be "natural born citizens" (google it yourself, don't take my word for it.)"
Does that mean something to you?