Great! Unfortunately, OSM is more complicated than you might think at first.

From osm.wiki/DE:Tag:leisure%3Dhorse_riding:

For a riding arena use leisure=pitch + sport=equestrian
...
The tag leisure=horse_riding is used to tag an equestrian facility, often called equestrian centre, riding centre or riding stables, where people practise horse riding, typically in their spare time. The tag is for the whole area which may contain one or more riding arenas, stables, riding halls, other related buildings or paddocks (compare also Riding).
...
Use this tag only once for the whole facility, don't use it multiple times on different objects belonging to it (one feature, one OSM element rule

On Bing it looks like way/1282375878 is a riding arena, like way/1263039130. The other areas could be paddocks or something similar.

Hi,
way/1282375878, way/1282375879 and way/1282375881 are no riding centres. way/962996369 is an example for a riding centre. Please have a look at osm.wiki/DE:Tag:leisure%3Dhorse_riding

Sorry, this was the wrong link. I meant way/1282431308

Hi,
way/128243130 looks like a riding arena and not like a riding centre on Bing etc. Compare way/1004939152 and OSM wiki.

Auf der Seite osm.wiki/DE:Riding kann man was finden:
osm.wiki/DE:Tag:landuse%3Danimal_keeping

Leider gib es dafür in iD (und auch sonstwo) noch kein Preset; also muss man es im Wiki finden und eintippen. Daher kommt es auch öfters vor, dass Leute Paddocks als Reitplätze taggen.

Was passiert denn auf dieser Fläche wenn da nicht geritten wird? Gebäude stehen anscheinend nicht drauf und für einen Reiterhof/Reitstall/eine Reitanlage ist die Fläche auch ziemlich klein; vgl. osm.wiki/DE:Tag:leisure=horse_riding.. Ist es ein Paddock für Pferdehaltung?

Hi,
You have removed the building tag from way/273996583, Now it is no longer rendered as a building. Isn't it a building? The tag leisure=horse_riding should be used for the whole facility and not for some building, compare
leisure=horse_riding
way/442937025
node/8755586716

Hi,
warum hast du denn way/667759376 umgetaggt? Das sieht auf den Luftbildern genauso aus wie way/1281435093, also wie ein Reitplatz (und nicht wie eine Reitanlage). Rätsel, Rätsel.

Hi,
riding arenas and paddocks are no riding centres (leisure=horse_riding). Please use the correct tags, e.g. leisure=pitch + sport=equestrian for arenas and landuse=animal_keeping + ... for areas that are nor for riding. Compare
osm.wiki/How_to_map_a#Riding_centre_(equestrian)
way/595858874

Traduzione automatica:
Ciao,
le arene e i paddock non sono centri di equitazione (leisure=horse_riding). Si prega di utilizzare i tag corretti, ad esempio leisure=pitch + sport=equestrian per le arene e landuse=animal_keeping + ... per le aree che non sono destinate all'equitazione. Confronto
osm.wiki/How_to_map_a#Riding_centre_(equitazione)
way/595858874

Tradotto con DeepL.com (versione gratuita):
Ciao,
le arene e i paddock non sono centri di equitazione (leisure=horse_riding). Si prega di utilizzare i tag corretti, ad esempio leisure=pitch + sport=equestrian per le arene e landuse=animal_keeping + ... per le aree che non sono destinate all'equitazione. Confronto
osm.wiki/How_to_map_a#Riding_centre_(equitazione)
way/595858874

Hi,
You have retagged a building to a riding centre. Now this building is no longer visible as a building on the map. Also the area of the riding centre is probably larger than the area of this building (see https://www.centro-ippico.net/cecinese.php), If the building is a riding is for horse riding you can use building=riding_hall for it. I would recommend to create a separate object (node or area) for the riding centre, compare leisure=horse_riding and way/268779943#map=17/43.64313/10.38555

Hi,
way/1280970869 looks like a riding arena. A riding arena is not a riding center and should be tagged like way/1257476842, compare osm.wiki/How_to_map_a#Riding_centre_(equestrian)

Hi,
die Bedeutung des Tags für "Reitanlage" (leisure=horse_riding) ist NICHT: "dieses Ding gehört zu einer Reitanlage", sondern "Eine Anlage, auf der Menschen in ihrer Freizeit reiten, oft als Reitanlage, Reitstall oder Reiterhof bezeichnet. Die Anlage sollte das ganze Gelände einschließen:. ... Verwende das Tag nur einmalig für die gesamte Reitanlage, verwende es nicht mehrfach an verschiedenen zugehörigen Objekten ...", siehe osm.wiki/DE:Tag:leisure=horse_riding.
Reitplätze, Longierzirkel oder Reithallen sind keine Reitanlagen und sollten NIICHT mit leisure=horse_riding getaggt werden, siehe Wiki. Beispiel: way/27331826#map=17/48.65251/10.81816&layers=P

That looks correct to me. Instead of osm.wiki/Tag:website=... you could use osm.wiki/Tag:contact:facebook=... here, see contact:facebook=*

Hi,
way/1280022172 looks like a riding arena (similar to way/1170874739), but you habe tagged it as a riding centre (leisure=horse_riding), compare osm.wiki/Tag%3Aleisure%3Dhorse_riding

See also previous discussion: changeset/150939363

Hi,
way/1280195200 is not a riding center, it is just a riding arena and therefore should be tagged leisure=pitch + sport=equestrian. In iD there is a different preset for riding arenas that sets these tags. Compare way/707965571 and osm.wiki/How_to_map_a#Riding_centre_(equestrian)

Hi,
was ist denn way/1279853115? Auf den Luftbildern kann ich nichts sehen, aber von der Größe her vermute ich, dass es wahrscheinlich ein Reitplatz ist wie z.B. way/410114894. Sollte dann anders getaggt werden, vgf. osm.wiki/DE:Tag:leisure=horse_riding

Hi,
way/519138147 is not a riding centre (leisure=horse_riding), it is just a riding arena and should be tagged leisure=pitch + sport=equestrian .Compare osm.wiki/How_to_map_a#Riding_centre_(equestrian) and way/436845393 . In JOSM you can use the preset for sport=equestrian.

Most (80%) buildings are just tagged building=yes, see https://taginfo.openstreetmap.org/keys/building#values