OK, it's the first time I've looked at names, and I felt, similarly to you, that it was backwards. 

As for the boundaries themselves, the first link shows land ownership, so it would only be appropriate to import that data into OSM if you've got evidence that access is directly linked to ownership. Is that the case?
What seems to be clear though, is that it's not a `leisure=nature_reserve`, and would likely be best represented by updating the existing `natural=forest` polygons into an MP with the appropriate access and operator tags added. Much as:
https://www.openstreetmap.org/relation/3155570 or https://www.openstreetmap.org/relation/1849496