There is probably all sorts of case law that broadens the meaning of “public road” to be anywhere the judge wants it to be. Especially your own driveway, but also parking lots.
Public road has a legal definition in any state. I can drink from the bottle and drive around my yard all I want, and I can let my 8 year old drive, too, subject only to child endangerment laws. Unfortunately, the automobile and the coming of public roads, instead of horse trails, gave government an excuse to put all kinds of conditions on people who traversed those roads. Those laws don’t apply on private property. Whether they apply to publicly owned lands, like a rink, would depend on how the law of that state defined a public road for purposes of licenses, DUIs, speeding, etc.