Hi,
https://www.openstreetmap.org/way/1320020873 is surely not a horse riding centre; it is too small for that. It is probably a round pen, and round pens are usually tagged like riding arenas: leisure=pitch + sport=equestrian. Similar case: https://www.openstreetmap.org/way/1234386940/history
Also https://www.openstreetmap.org/way/1320020872 does not really look like a horse riding centre; but I am not sure from the imagary what it is. Anyway, leisure=horse_riding "... is for the whole area which may contain one or more riding arenas, stables, riding halls, other related buildings or paddocks" (from https://wiki.openstreetmap.org/wiki/Tag:leisure=horse_riding), like here: https://www.openstreetmap.org/way/1319870274