Searching online for the phrase "Handy Cross Hub Access Roads", it appears to me that Buckinghamshire Council is treating all roads in the Handy Cross area - near the leisure centre, the supermarket, the hotel, etc - as being one large "off-street" area.
When driving from the nearest main road (Marlow Hill) to the leisure centre, I see that the council has a small black/white sign stating that restrictions and penalties apply, implying that the apply past this point. GSV link with signs on either side of the road, although the quality makes it difficult to read the signs:
Find local businesses, view maps and get driving directions in Google Maps.
Please tell us about the family member you were escorting to the leisure centre, because you were obviously dropping somebody off. If the family member is classed as "vulnerable", (child or elderly infirm person etc), you were engaged in what is termed "assisted boarding/alighting", which should get you your PCN cancelled.
They are a child, but I have doubts about whether this would apply. I left the car to walk them into the building; it wasn't for long, but it was more than alighting from the vehicle.
This looks like nonsense - it can't be restricted if it allows drop-offs and as you say there are no terms establishing the area as an off-street car park.
Having looked online here and there in the last 24 hours, this is something I wondered also. My understanding is limited, but I had thought that Contravention 81 ("Parked in a restricted area in an off-street car park") only applies when
no-one can use that area. The signs make clear however that
some vehicles
can use it: coaches, minibuses and taxis. In other words, surely the contravention at play should have been one that reflected the fact that my vehicle isn't one of the permitted types, rather than the fact that it's a "restricted area"...?