Actually, I don't think that the lake is on private property. It's located in the City of Bay Lake, which is controlled by the Reedy Creek Improvement District (RCID). While RCID essentially exists to benefit Walt Disney World, it still technically is a municipality.
Regardless, I'm not sure that matters. According to the link that hrhdhd provided, exceptions only exist for fish ponds of 20 acres or less. Apparently, a fish pond is "is a man-made pond constructed for the primary purpose of fishing." Since Bay Lake is a natural lake, it wouldn't qualify.