Hi,
https://www.openstreetmap.org/way/1248958378 and https://www.openstreetmap.org/way/1248958418 are riding arenas and not riding centers. Riding arenas are usually tagged leisure=pitch instead of leisure=horse_riding. In JOSM you can use the preset for sport=equestrian to tag them correctly.

The paddock (https://www.openstreetmap.org/way/1248958381) shoud not get a horse_riding tag either. The tag leisure=horse_riding could be used to map the whole "horse property" as a separate node or area; example: 
https://www.openstreetmap.org/way/773021652
See also: https://wiki.openstreetmap.org/wiki/How_to_map_a#Riding_centre_(equestrian)