So you are saying that because their parents "naturalized" and became citizens, their offspring were therefore citizens? And Here I thought you couldn't learn anything. :)
In your example, it says that you have to live in New York to be a resident (citizen) of New York. Just like today. It says nothing about US citizenship though. So someone passing through New York when born is a US citizen but becomes a legal resident in the state where his parents eventually take him back to where they live. In your example, if that child of a transient alien was born in a neighboring state (thus becoming a US citizen) and then settled to live in New York, he would be a New York citizen according to section 5.2
What you may not know is that Law from New York which I cited, was created as a result of the legislature's disagreement with the New York State court in Lynch v Clarke which ruled that because New York had no law on the subject, they would use English Common law to decide that anyone born in New York was a citizen.
As a Result, the New York Legislature decided they needed a law defining it so that the courts wouldn't try to use that English Common law crap. In creating the law, they specifically excluded the Children of Transient Aliens. In those days, In order for one to be a Citizen of the United States, you also had to be a Citizen of a State. (DUH.)
The Congress could set requirements for someone to be NATURALIZED, but the States decided who would be born citizens within their own borders, and Native-born/Natural-born citizenship was derived from that. If New York didn't recognize the Children of Transient Aliens as a citizen of their State, they had no basis to claim American citizenship either.
Even if you were recognized as a citizen of a state, that did not necessarily mean you were considered an American Citizen. Here is an example from the October 10, 1811 edition of The Alexandria Herald newspaper. This article was written by James Madison regarding a "James McClure" who was BORN in Virginia, but not considered a Citizen of the United States.
If your understanding of "born on the soil" = "natural born citizen" then why would James McClure's citizenship be in doubt in 1811? (A time a lot closer to the Founding era, and during which they ought to know the truth one way or the other.)