Hi,
https://www.openstreetmap.org/way/1368108899 is not a horse a riding centre. It is probably the outdoor arena described on https://www.arkenfieldstables.co.uk/facility-hire. In OSM such riding arenas are usually tagged with leisure=pitch + sport=equestrian.

Maybe https://www.openstreetmap.org/way/1368108902 is a small riding arena too; I don't know. Or perhaps it is something else (paddock?).

You can use the tag leisure=horse_riding for the whole riding stables / equestrian centre; see https://wiki.openstreetmap.org/wiki/How_to_map_a#Riding_centre_(equestrian), example: https://www.openstreetmap.org/way/380370797

I don't know what https://www.openstreetmap.org/way/1366053178 is, but I think it cannot be a riding arena because there cannot be buildings on a riding arena.