At what point does a body of water qualify as a "lake"? There must be a thousand or more private bodies of water in Texas under 100 acres. If Fairfield land is under 1 ownership, and the park is a lease that can be terminated, I guess that makes it private, whether current access is allowed or not. Same as a stock pond.
I have wondered this myself in the past. According to my internet searches, there appears to be no size criteria for a lake.