I don't have time to drive to Saugerties just to check if a small is named correctly, especially considering the possibility that it wont even have a name sign labeling it, but what I can tell you is that the properties on this road are listed as on Shag Bark Pond Rd, and this US census data map also confirms the road's name. It states the names are up to date for 1990, which is a while ago but there is no reason to believe the name has changed. NYDOT doesn't have the road listed in their 2022 data on local roads in Ulster County, but neither does it describe the road by another name.

https://www2.census.gov/geo/maps/blk1990/st36_NewYork/36111_Ulster/90B36111_012.pdf