I finally found the baptism record for Mary M. Cashow, who was stated in that family story book to have been born in "Rhinebeck, Long Island."
She was baptized on July 7th, 1793 at St. Paul's Lutheran Church of Wurtemburg (about 5 miles from Rhinebeck), Dutchess County, NY. The record states she was born on May 16, 1793, but doesn't say where.
I guess it's possible that she was born on Long Island (presumably Queens County) in May, and then for some reason the family went to Dutchess County, and she was baptized there in July.
Question is, why would they have done that? The war was over and I know of no family connection to Dutchess County (no siblings born there, etc).
Funny how some records, when found, only lead to more questions!