The record referenced is just an index to the vital records.
Without access to the Manchester records it is difficult to be sure. The birth was registered in Manchester but in that time period, Vermont towns typically required a family to register all births (including location) after they arrived. An example in my line is
https://www.familysearch.org/ark:/61903/3:1:939V-3F33-6Z?i=3336&cc=1784223 which would lead you to believe born in Woodstock, Vermont but he was born in Middleborough, Massachusetts. The mother's residence might indicate that was the birth location but there is no consistency in the index cards that would make it definite. I would lean toward the New York location.
I don't see Manchester records on FamilySearch for that time period and New York is just a problem all around. You could narrow it down if you can find a first reference of living in Manchester.