Hi,

the 'Val du Bois des Bégines - Begijnenbosdal' is not the one next to the Military Hospital but this one nearby: https://www.openstreetmap.org/way/46869624#map=17/50.90116/4.38876&layers=N

There is a sign at one of the entries of the wood, on 'Petit Chemin Vert - Korte Croenweg', that clearly states it: https://www.mapillary.com/app/?lat=50.90105&lng=4.3920861111111&z=19.9&pKey=899259213953908&focus=photo&x=0.6713762203419813&y=0.46827123658566305&zoom=1.6138550750656386

If you have got the information from gardens.brussels website, they are wrong here.
I don't know if the wood near the Military Hospital has a name.

I undid your change here.

Cheers