1. I have no idea how competent New York State Historical Society is in dating rugs - they are no rug dealers.
2. Experianced dealers have seen a lot of rugs and know their age. So they can tell from design, color and structure what age they are (approximatley). As RugPro posted before, it should be checked by an expert. I can give you an idea about age from the picture, but this has absolutely nothing to do with a personal examination of the object. the more information the person who is inspecting the rug gets, the closer will be the dating of the rug. Antique persian rugs are art. Like with paintings, a photo of it can give an idea, nothing more!
3. As I posted, I think it was made 1900 +/- 15 years. This is based on the information I can get from the posted pictures. If you pay 2000$ for the rug it is a very good price if the rug is not dry or rotten, or pile is very low.
4. I recommend you take a few bucks in your pocket to pay an expert to check what you bought. Don`t tell him the story of 1790 and where ever the rug came from. Just let him look at the piece and give you his view of the rug.
5. If the rug was in good shape I would buy it for 2000$ with age 1915. So - as long as the condition of the rug is good you made no mistake.
I checked all available photos on the web showing historic photots of Schuyler Mansion in Albany - there are quite nice rugs to see, but I didn`t see the one you bought. So maybe this was added later.
If you look at the history of the building:
After Philip Schuyler's death in 1804, the land comprised over one hundred building lots which were divided among his numerous children.[4] From 1886 to 1913, the mansion served as an orphanage until the state assumed ownership.[9] It was restored and dedicated as an historic monument on October 17, 1917. (source: Wikipedia)
Knowing this history, it is clear that most furniture must have been removed 1886 and was newly furnished after 1917. I don`t think they spend a fortune buying rugs that where more than 120 years old at this time - they probably took what they where able to get to furnish the house.