Our field is privately owned and leased by the club. Access to the property is by membership and the public is welcome, but they have no "rights" as far as access is concerned. The club, and it's members have contractual rights of access, not the general public.
There are several flying fields around Kansas City that are on public ground, city parks, etc. The general public has complete access to the facilities as long as they comply with the safety rules, etc.
In the past, I have flown at closed fields that are by invitation only, and as the name indicates, are open only to invitees.
As to which case applies to the OP, there are not enough facts present to make an accurate determination. In any event, the AMA really has no jurisdiction in all cases as they don't own the land, and that is the real issue.