The Founders and Framers made their views clear in the Naturalization Act of 1790, signed into law by President Washington:
“the children of citizens of the United States that may be born beyond Sea, or out of the limits of the United States, shall be considered as natural born Citizens.”
And in today’s parlance, current U.S. law states that one of the criteria for being a Citizen of the United States at Birth is:
“a person born outside the geographical limits of the United States and its outlying possessions of parents one of whom is an alien, and the other a citizen of the United States who, prior to the birth of such person, was physically present in the United States or its outlying possessions for a period or periods totaling not less than five years, at least two of which were after attaining the age of fourteen years...” -US Code Title 8 Section 1401
And for some peculiar reason the words "shall be considered as" are invisible to you guys.
If anything, those words are a tacit admission that the two things are different.