So are you asserting that there is a difference, legally, between being a "natural born citizen" and "born as citizens" or "citizens at birth"?
Do you have a scholarly legal reference (such as a court decision or a legal textbook, such as Blackstone or Vattel or the like) that draw that distinction?
Because, to my knowledge, in this country, there are two types of citizens: those that are born that way (i.e., natural-born) and those who become citizens at some point later in their lives (i.e., natural-ized)
Yes, why else would they have left out "natural born" in all laws subsequent to the 1790 law, as your post of the laws through the 1802 law shows?
It has to be that way. Congress only has the power to define a Uniform Rule of Naturalization. Thus any law on the subject of acquiring citizenship, at birth, en mass or by individual application, must be concerned with "naturalized" citizenship, otherwise Congress would have been exceeding its delegated powers, and the law would be unconstitutional.
The 14th amendment makes all persons born in the United States "citizens", but the authors of that amendment also left off "natural born", thus leaving us to look at contemporary writings or writings known to be relied up by the founders, for the definition of Natural Born as used in the Constitutional requirements for eligibility to the office of President.