Question: What is the village of the town, with the same name, in Saratoga County, New York near a lake?

Context paragraphs:
Saratoga, New York: Saratoga is a town in Saratoga County, New York, United States.  The population was 5,141 at the 2000 census.  It is also the commonly used, but not official, name for the neighboring and much more populous city, Saratoga Springs.  The major village in the town of Saratoga is Schuylerville which is often, but not officially, called Old Saratoga.  Saratoga contains a second village named Victory.

New York State Route 9P: New York State Route 9P (NY 9P) is a state highway in central Saratoga County, New York, in the United States.  It is an alternate route off U.S. Route 9 between Malta and Saratoga Springs that connects US 9 to the eastern shore of Saratoga Lake.  NY 9P meets NY 423 in the town of Stillwater and Interstate 87 in Saratoga Springs.  In its final mile, NY 9P serves the Saratoga Race Course.

Stillwater, New York: Stillwater is a town in Saratoga County, New York, United States.  The population was 7,522 at the 2000 census.  The town contains a village called Stillwater.  The town is at the eastern border of the county, southeast of Saratoga Springs and borders both Rensselaer and Washington counties.  Saratoga National Historical Park is located within the town's limits.  There is a hamlet in Minerva, Essex County, New York with the same name which has nothing to do with this town.

Saratoga Mall: Saratoga Mall was an enclosed shopping mall in Wilton, New York near the city of Saratoga Springs, New York.  It was demolished in 1999.  It was previously known as Pyramid Mall Saratoga and was located on Route 50 just off Exit 15 on I-87 (the Adirondack Northway portion).  After demolition, it was replaced by a big box strip center known as Wilton Square.

Schuylerville, New York: Schuylerville is a village in Saratoga County, New York, United States.  The population was 1,386 at the 2010 census.  The village is named after the Schuyler family.  Schuylerville was the site of the Battles of Saratoga (1777) and contains several historic buildings including General Schuyler House and Old Saratoga Reformed Church.  The Saratoga Monument is located in nearby Victory, Saratoga County, New York.  Schuylerville High School and a library are part of the village.

Waterford, New York: Waterford is a town in Saratoga County, New York, United States.  The population was 8,515 at the 2000 census.  The name of the town is derived from its principal village, also called Waterford.  The town and village are in the southeast corner of Saratoga County, and north-northwest of Troy, New York.  It is located at the junction of the Erie Canal and the Hudson River.

Ballston Spa, New York: Ballston Spa is a village in Saratoga County, New York, United States.  The population was 5,556 at the 2000 census.  It is the county seat of Saratoga County.  The village is named after Rev. Eliphalet Ball, a Congregationalist clergyman and an early settler.  The village lies on the border of two towns and is partly in the Town of Ballston and partly in the Town of Milton.  The village is southwest of Saratoga Springs.

New York State Route 423: New York State Route 423 (NY 423) is an east–west state highway located within the town of Stillwater in Saratoga County, New York, in the United States.  It extends for 3.86 mi from an intersection with NY 9P on the eastern shore of Saratoga Lake in the hamlet of Snake Hill to a junction with NY 32 southwest of Saratoga National Historic Park.  The road was acquired by the state of New York in the early 1920s and designated as NY 423 in the early 1930s.

Galway Lake: Galway Lake is located west of Galway (village), New York, in the town of Galway in Saratoga County, New York, United States.

Galway (village), New York: Galway is a village in the town of Galway in Saratoga County, New York, United States.  The population was 214 at the 2000 census.  The village is north of Schenectady.  Galway Lake, a vacation area, is west of the village.

Supporting paragraphs: Galway Lake Galway (village), New York
Answer: Galway