Hi,
https://www.openstreetmap.org/way/1323771239 looks like a riding arena and not like a riding centre (leisure=horse_riding). In OSM riding arenas are usually tagged leisure=pitch + sport=equestrian, examples: https://www.openstreetmap.org/way/607473493 and https://www.openstreetmap.org/way/1116841118; compare https://wiki.openstreetmap.org/wiki/Tag:leisure=horse_riding.

In JOSM you can use the preset for equestrian sport to tag them.