text: seeing that this is obviously an area open to the public, for family outages and other tourism, i'd tend to be a bit lenient with the osm rules and allow things in the name field that very likely appear as labels, whether on a map available to visitors, guideposts, or marking the objects themselves.
  a greenhouse and amphitheatre are typically rendered no different than any other building, but i'd be sure there is some signpost  pointing to them, and that is probably how they are referred to lacking a formal name.
  
  given that the target audience may well be using a deficient app, within the confines of this public space, just like at disneyland/-world  where naming goes similarly overboard, having a descriptive name in use will help to locate the particular garden or trail, without turning to a particular renderer.
  
  outside this public area, not so much.  however, one thing to occur to me recently is that to a non-native, their app may display generic descriptions in their own language, not matching the posted terms, so to display the native language, even for herb garden, will match what is very likely signposted in some way.