For those who are confused, New York has a different system of political subdivisions than other parts of the country. Each county is divided into Cities and Towns (or townships). All land in a county falls into either a City or a Town. Suffolk County has no cities so the county is divided into "Towns" which include settled area and open space. We are primarily interested in the Towns of Babylon and Islip.
Within "towns" there can be Villages, which are settled areas that are organized municipalities that provide various services. Within Towns there can be "Hamlets" that are nothing more than names people call the area. Hamlets have no legal status; all services are provided by the Township.
It is possible for a Hamlet or Village to fall within two towns. In the Towns we are primarily interested in, the Hamlet of West Islip is partly in Babylon and partly in Islip. There is a Village of Babylon in the Town of Babylon and a Village of Islip in the Town of Islip. This sometimes causes confusion. It is not always clear whether the Town or Village is being referred to.
Because Hamlets have no "official" boundaries, there is often no general agreement on "name" to call a specific area. Neighbors may use different names to describe where they live.
The CVS in question appears to be on the Montauk Hwy, in the Hamlet of West Islip, in the Township of Babylon but they appear to refer to the store as the as the "Oak Beach" CVS.