https://www.cavallinoazzurro.com/attivit%C3%A0equestre states that they offer riding lessons for advanced riders (not beginners), but are not purely a riding center. Therefore, I agree that leisure=horse_riding is probably inappropriate. There is a tag https://wiki.openstreetmap.org/wiki/Tag:social_facility=equine_assisted_centre but this is probably also  inappropriate.. Maybe https://wiki.openstreetmap.org/wiki/Tag:guest_house=agritourism fits (?).

The tag tourism=trail_riding_station was invented (long ago) for places where horse ridiers can stay overnight on longer treil rides (see https://wiki.openstreetmap.org/wiki/Proposal:Trail_riding_station#Rationale). But this is not well explained on the wiki page and I have seen other mappers use this tag for different places. https://wiki.openstreetmap.org/wiki/Tag:landuse=animal_keeping may be a better tag for your "horse-walking area". But there is no preset for this in iD yet; you would have to type these tags in.

As you see, OSM is still lacking many things and difficult to understand. Sorry.