Hi,
These small round pens like 
https://www.openstreetmap.org/way/1185756798
and
https://www.openstreetmap.org/way/1185163885
are no riding centres (leisure=horse_riding); they should better be tagged leisure=pitch + sport=equestrian. See
https://wiki.openstreetmap.org/wiki/Tag%3Aleisure%3Dhorse_riding

To find the preset for tagging riding arenas in JOSM search for "equestrian".