Try Rogers v Bellei
We thus have an acknowledgment that our law in this area follows English concepts with an acceptance of the jus soli, that is, that the place of birth governs citizenship status except as modified by statute.
Jus soli means "born on the soil" (usual diplomatic etc. exceptions) = natural born.
I like to cite Rogers v Bellei myself. It proves beyond a doubt that a "citizen at birth" is not the same thing as a "natural born citizen."
In any case, New York, California, et al have "modified the statute", so that place of birth doesn't apply to the children of transient aliens. Here is a copy of the New York law. (1845)
Now this seriously begs the question. How can someone be a "natural born citizen", if they are explicitly BANNED by State law from even being a citizen?
You realize that code was blown away by the 14th Amendment?
And that American citizenship was a federal issue, not a state issue? If this law had been challenged it would have been deemed unconstitutional. However, that did not happen because of the Civil War and was then rendered moot by the 14th Amendment.
Your NY Code simply documents the fight over how states and the federal government would share power. Before the 14th amendment states had a say in who was a citizen and what rights they had. The states have not had a say in citizenship since the 14th.
Since what you have posted (code from 1845) has since been changed by Constitutional amendment and statute, it has no applicability to a birth in 1961. Not to mention that it applied only to New York state, not any other state. Are you claiming 0bama was born in New York?