I saw that you added that, and thank you, it is relevant and does clarify that these are closed areas.  That's where I have the disconnect:  a closed area really isn't a "leisure" anything, which is why the leisure=nature_reserve tag seems inaccurate to me.  When I use that tag, it implies that there is public access (though limited to staying on trails and/or the "wildlife viewing area" which is often fenced, has interpretive signage and suggests binoculars).

These (closed) areas really are not that.  The other tags, withOUT leisure=nature_reserve, are sufficient to express these.  Although the protect_class=5 that causes rendering in Carto is often "over-applied" (and not strictly an IUCN denotation).

I've been active for almost 15 years in trying to improve the tagging on these in OSM, and it's been a struggle, with a lot of misunderstanding of existing tagging, and more difficult, tagging which evolves.  For example, Brian Sperlongano (Zelonewolf) and Kevin Kenny (ke9tv) and I have been working to reduce the protect_class numbering (which is a mess) and introduce "plain English tagging" with protection_class.  I realize it can be difficult to follow our wikis, newer proposals, the "drift" in semantics over time as tagging conventions change and evolve, but please take a look at as much wiki as you can on these topics:  it's complex, but it does sink in after seeing the historical trajectories OSM has taken in the last decade or so.

There is a leisure=nature_reserve in Santa Cruz (Pogonip Open Space Reserve) which I believe is accurately tagged (that), especially as it is an "open" area, not closed to the public.  These (MROSD, others around the Bay Area) which have closed areas shouldn't have leisure=nature_reserve applied.  Thanks, I'm happy to further dialog, offer perspective and answer any questions, although I certainly don't know everything!