I guess Keep off the grass
can be a traffic or road sign if you've got cars running around the park! (Only kidding, my friend.
Yes, Keep off the grass
is a public notice, I suppose, but we normally use the word notice
to mean the kind of sign that imparts information of some sort to the public.
For example, I stop by the City Hall in the town where I live. On the bulletin board enclosed in glass to the right of the front entrance, there's a sign that says, "Hearing on request for new zoning laws. Tuesday, May 27 at 7:00 p.m."
That's what I consider a public notice. The government is telling me what they have scheduled and that I can attend the hearing if I choose to.