Hi,
https://www.openstreetmap.org/way/1234029521 is a riding arena and not a riding centre. iD has a separate preset for riding arenas which sets the usual tags leisure=pitch + sport=equestrian (instead of leisure=horse_riding). Compare https://wiki.openstreetmap.org/wiki/Tag%3Aleisure%3Dhorse_riding
https://www.openstreetmap.org/way/1203406685 (example)