Re: Ballston Lake and Ballston Spa
The genealogists "around 1900" probably were talking about the Town of Ballston.Within the Town of Ballston are Ballston Lake, and a part of the incorporated Village of Ballston Spa.(The largest physical portion of Ballston Spa is in the Town of Milton.) It is possible that someone used "Ballston" as an abbreviated form of "Ballston Spa" but I would think it more likely that they were being accurate as far as the physical location and stated Ballston meaning within the Town of Ballston.
It's easily enough checked with census records, tax records, land records and the like.