What follows is from the Smithfield Historical Society, through the Smith Appleby site. I found it to be more clear in understanding the layout of the villages within municipalities that were at one time Smithfield prior to 1871.
South of the Blackstone River, Central Falls, Lincoln and North Smithfield were all part of Smithfield which was incorporated 1730; RECORDS PRIOR TO 1730 ARE IN THE PROVIDENCE CITY ARCHIVES.
This is why the place of birth and place of burial may today be located in one of the other municipalities and not in Smithfield. The Smithfield site points out how one could be born at one house and die at the same house yet their record of birth and their record of death could be in separate towns.
(Lends new meaning to the term "lost soul"?)
When trying to figure out villages....
CENTRAL FALLS Moshassock
LINCOLN Albion Lime Rock Londsdale Manville Saylesville Saylesville-Heights Quinnville
NORTH SMITHFIELD Branch Village Forestdale Primrose Union Village Slatersville Waterford
SMITHFIELD Esmond Georgiaville Greenville West Greenville Mowry's Corners Spragueville Stillwater
WOONSOCKET South of the Blackstone River Globe Village Hamlet
As the Smithfield site informed me: Smithfield was divided in 1871. (North Smithfield was originally incorporated as the Town of Slater.) Also, a portion of land was deeded over to Woonsocket. Central Falls remained a village in Lincoln when Smithfield was divided. Central Falls was considered "down town" and was where the Smithfield Town Hall was located. After the division, a new Smithfield Town Hall was established in present-day Smithfield. So any vital records, for Smithfield, that date between 1730 and 1870 are kept at Central Falls Town Hall.
Central Falls separated from the Town of Lincoln in 1895 and incorporated as its own municipality.