If the aerodrome is certified, the operator would need to demonstrate to CASA that they are taking steps to control access to airside areas.
Under common law, a person has a reasonable defence against charges of trespass if they are accessing a place open to the public that has no signs, fences, gates, or other notices or measures indicating that it is not a public place. If you landed at the airstrip and approached these notices from behind (ie from airside), and had consulted ERSA and NOTAM and the air/ground signalling area near the windsock, then you have established a solid defence against prosecution (unless you went on to cut the lock from the gate, in which case you have committed an entirely new offence).
Clearly, then, the signs are intended to keep the airport visiting aircraft-friendly by ensuring the airside area is for you to park on, not for Farmer Jack to store his hay bales or the local scout unit to hold an open day on or the local car club to do performance testing (all three of which I have seen at rural airstrips more than once!).
I would suggest that these signs, while poorly worded, are obviously not intended to apply to crew or passengers of an aircraft. As they're confusing, continue to call the council about them until they decide the cost of changing the wording will be worth it to stop the phone calls. But there's no need to become full of hatred for a council that, unlike others, seems to be trying to keep the airport as a place for aviation.