Doesn't the CLOSED bit just refer to whether the main gate is open or locked shut, and that the office building is closed?
The runway should remain serviceable
English humour right?
The aerodrome operator is morally if not also legally responsible for preventing the use of the runway if it is not "serviceable". Any number of reasons could be the cause, but closing property you own, is the privilege of ownership, and must be respected by everyone else.
As the pilot, you are at least honour bound to assume that the
entire aerodrome is closed, when "closed" is associated with it, unless the closure is more specific to be only the club house, parking lot or the loo.
I own my runway, and I sometimes close it (with "X's") when it is too soft. Also typical in Canada is the statement: "Limited winter maintenance". This should be interpreted as "the owner does not mind of you land, but assure that the runway conditions meet your expectation, 'cause there is no assurance. So, if you don't know, treat it as closed.
It's great for students, or any pilot for that matter, to fly from grass, and the more opportunities the better. But, that does not translate to "I'm gonna land, 'cause I see a runway in front of me". Assure you have whatever permission may be appropriate for the aerodrome.