What I said was that the first naturalization act exempted children born overseas or out of the US from needing naturalization by declaring them to be natural born citizens as long as their fathers had been at some time resident in the U.S.The first naturalization act, not unsurprisingly, was 100 percent about naturalization and had nothing to do whatsoever with altering or widening the understanding of the Constitutional phrase natural born Citizen. All it stated was that for the purposes of naturalization, the children of American citizens born overseas would "be considered as" (but not actually be) natural born Citizens -- that is they would not need to go through the naturalization process when they returned with their parents to the USA.
I suspect the reference to natural born Citizen was deleted altogether five years later because the founders knew that congress's powers over naturalization did not extend to redefining the meaning of the Constitution (not that that was ever their intent) and upon the consideration of time, felt that such language might be misleading and therefore had no place in a law about naturalization.
“Considering the history of the constitutional provision, the clause’s apparent intent, the English common law expressly applicable in the American colonies and in all of the original states, the common use and meaning of the phrase “natural born” subject in England and the American colonies in the 1700s, and the subsequent action of the first Congress in enacting the Naturalization Act of 1790 (expressly defining the term “natural born citizen” to include those born abroad to U.S. citizens), it appears that the most logical inferences would indicate that the phrase “natural born Citizen” would mean a person who is entitled to U.S. citizenship “by birth” or “at birth.” Such interpretation, as evidenced by over a century of American case law, would include as natural born citizens those born in the United States and subject to its jurisdiction regardless of the citizenship status of one’s parents, or those born abroad of one or more parents who are U.S. citizens (as recognized by statute), as opposed to a person who is not a citizen by birth and is thus an “alien” required to go through the legal process of naturalization to become a U.S. citizen.
The weight of scholarly legal and historical opinion, as well as the consistent case law in the United States, also supports the notion that “natural born Citizen” means one who is a U.S. citizen “at birth” or “by birth.” The Constitution of the United States of America, Analysis and Interpretation, notes that “[w]hatever the term ‘natural born’ means, it no doubt does not include a person who is ‘naturalized,’” and, after discussing historical and legal precedents and arguments, concludes that “[t]here is reason to believe ... that the phrase includes persons who become citizens at birth by statute because of their status in being born abroad of American citizens.”—Qualifications For President and the “Natural Born Citizen” Requirement, Congressional Research Service