I believe Thomas and Matilda were both born at Halfmoon, where they were living in 1829 when their first child, Mary Ann, was buried at Newtown Cemetery that September. They were still there in July of 1832 when the phrase "Thomas Vincent of Halfmoon", appeared on an indenture for land purchased in the town of Victory, New York where Thomas would die ten years later. Of course, I checked Halfmoon for a John Taylor and the census of 1800 shows one in town, not far from Jeremiah Vincent, and John Vincent [Sr.] who was Thomas' grandfather. I believe Jeremiah Vincent was a relative of my Thomas, probably his father's uncle*. Furthermore, John B. Taylor, besides living near Jeremiah, served under him in the Revolutionary War. In fact, Jeremiah Vincent was a witness on John B's pension application in 1833 and John B. acted as witness for Jeremiah when he applied for a pension. Also, John B. in his application states he had lived at Halfmoon since the end of the war so Matilda, whose birth took place in 1802, must have been born there.
Finding John B's Revolutionary War pension application was exciting but unfortunately didn't shed any light on the question of his children. It did however provide his birthplace as Dutchess County. I looked in vain for a widow's pension since it seems Rebecca outlived John, but the two probably weren't married before the end of the war in 1783, which at that time disqualified her as the widow of a soldier.
Find A Grave shows a burial that took place in Galway, Saratoga County, in 1839 for John B. Taylor and one for "wife" Rebecca in 1845, no photos of any headstone, but with dates. Since I trust Find A Grave about as far as I can throw them, as always I sought and found another source for West Galway Cemetery. As I suspected it had more information than Find A Grave, with a transcription of the actual inscriptions on the stone, and is likely more trustworthy.
The census of 1830 shows John B. still in Halfmoon, but I can't rule this1839 burial out even though it's not in Halfmoon. Galway is only twenty-five miles distant, and it's always possible a child of John and Rebecca lived there and took their parents under their roof in their declining years. I do wonder if Rebecca was John's second wife given the twenty-year difference in their ages indicated by the dates on their grave markers, but so far even her last name remains a mystery. Furthermore, looking at the pension ledger containing John B's pension, payments to him stopped in March of 1839 tending to confirm that is indeed him buried in Galway.
The next obvious step would be to get a copy of Matilda Taylor Vincent's death certificate, but that's already been done. The lines where her parent's names should be were left maddeningly blank. I've exhausted obituary searches with no useful information turning up, so I'm at a loss for now. I should mention there was a Reuben Taylor also living at Halfmoon early on, maybe he's my next search...
* I don't understand why, given all the evidence to the contrary, but there is disagreement in online trees as to the father of John Vincent, the husband of Mary Clements and father of Thomas Vincent. For some reason, most come down on the side of Capt. Jeremiah Vincent. There is absolutely no proof of this and none is provided in the trees. This is partly why I ordered his records from NARA, hoping something contained in them would settle the question-- which it has.
The records, being his wife Mary's approved application for a widow's pension, prove beyond a shadow of a doubt that my fourth-great-grandfather, Private John Vincent of Halfmoon, died of disease in February of 1814 at Plattsburgh during the war of 1812, seven long years BEFORE Capt. Jeremiah made his will in 1821 in which he named his living son John I. Vincent an executor of his estate. That, taken with the 1817 guardianship of John's children, granted to Mary and her brother John Clements, rules my John out as the Captain's progeny.