How should "Local purpose (landscape and environment protection)" be tagged?

Not rendered, at least by Mapnik CSS on openstreetmap.org, but I really prefer to use landcover=* for this type of situation. I may not know what the use is (ornamental, storm water filtration, recreational, etc.) but I can see the area is covered with grass or trees or scrub or . . .

Alas, since landcover is not as widely used or rendered, I end up "tagging for the render" and use landuse= in addition to my preferred landcover=.

This land is grassy and looks to be regularly mowed,

If used for nothing else (eg not a landuse=recreation_ground), landuse=grass suffices. We would like to describe what's on the ground foremost. You could try tagging the official classification/purpose somehow. Perhaps designation=, if you want something more than description=.

with many trees.

Now this depends on how "many trees" there are. By the sound of your question, it doesn't seem that many, so you can leave it to the next step.

The example by @Kovoschiz demonstrates that a particular tag has radically different meanings in different parts of the world.

In the UK a recreation ground is close to a park but predominantly dedicated to pitches for team sports. It is not used to mean any area which is predominantly used for sport or leisure activities. In the US, on the contrary, the tag is used with a generic meaning (e.g., for ski areas), but also for patches of what in Britain might be described as amenity grassland or urban commons.

In the UK and elsewhere there is a tendency to use landuse=grass for this type of feature, but again substantial ambiguity exists. Both values will be rendered as areas of green, which may be adequate for your purposes, but these tags create problems for any type of analytical use of the data.