Never heard of OMS Open Map before. Is that something new or what do you mean. Maybe this table can help:
osm.wiki/Raster_tile_providers

Hi,
relation/18808715 should probably better be tagged leisure=sports_centre + sport=horse_racing, see osm.wiki/How_to_map_a#Racecourse_(horse_racing)

Maybe the whole site, including the surrounding buildings, is an equestrian centre, I don't know; example: way/863798326#map=18/39.667668/2.947887&layers=P

But the area way/1366005685 alone looks more like an outdoor riding arena. Riding arenas can belong to an equetrian centre or not. The current name "Caballos" does not seem to be a name of an equestrian centre.

If you search for "pista ecuestre" in iD you can chose from
* Pista de carrera ecuestre
and
* Pista ecuestre / Arena de rodeo
The first one is for horse racing, this is wrong here; the second one is what I think could fit here.

See also osm.wiki/How_to_map_a#Riding_centre_(equestrian)

Hi,
way/1366005685 looks like an outdoor riding arena and not like a riding centre (centro equestre, osm.wiki/ES:Tag:leisure=horse_riding). In OSM riding arenas are usually tagged with leisure=pitch + sport=equestrian, like this one: way/1131292483

I assume the whole Stables in Cressage is larger than just way/1365260718. At least the stable building also belongs to it. Compare
osm.wiki/How_to_map_a#Riding_centre_(equestrian)
way/1089752556#map=19/53.466954/-2.566878&layers=P (example)

Hi,
why did you retag way/1069695518 from pitch to horse_riding? It is riding arena and not a riding centre. In OSM riding arenas are tagged leisure=pitch + sport=equestrian, see osm.wiki/How_to_map_a#Riding_centre_(equestrian) . Also it makes no sense to have a small riding centre mapped inside of a larger one (way/1159491930).

I created https://github.com/streetcomplete/StreetComplete/issues/6155

Hi,
way/1365260718 is quite small for a horse riding centre. It may be a small horse riding arena. In OSM horse riding arenas are usually tagged as pitch + equestrian, example: way/1192427748

Hi,
eine Pferdekoppel ist aber keine Reitanlage. Hier ein Beispiel für eine Reitanlage: way/694107108
Eine Pferdekoppel kann man mit landuse=meadow + meadow=pasture taggen, siehe osm.wiki/DE:Riding

Hi,
you have added leisure=horse_riding to way/39979940, but this building belongs to Club Hippique des Alpes (way/153093821) which was and is correctly tagged with leisure=horse_riding. To map a small equestrian centre inside of larger one makes no sense to me and is a violation of the one-feature-one-osm-element rule. Maybe you can use building=riding_hall instead.

Hi,
way/1364865247 does not look like a horse riding centre, it looks more like a horse riding arena. In OSM horse riding arenas are usually tagged leisure=pitch + sport=equestrian, like here: way/1069062624
See also: leisure=horse_riding

Done in
changeset/163254860
and
changeset/163255503

@Allison P:
I wrote to the DWG last night already and asked them to take a look at this discussion. (That was the first time I ever wrote to them.) That's why I first thought that's why you commented here. But if you're not one of them, someone else from DWG may show up here anyway,

Thank you for fixing.

You have mapped not 6 but 11 horse highlines and tagged them as riding centres (and once you have mapped a riding centre correctly: node/12496714162). And you only started with OSM 2 months ago. How are you going to continue? Should the whole of the USA be covered in incorrectly tagged horse highlines in a few years' time? The more often a tag is used incorrectly, the more useless it becomes. I don't know of any tag that is intended to make any text appear anywhere on the maps. That is not the principle of how OSM works. The notes are readable for humans, but cannot be processed automatically.

If you want to see horse highlines in OsmAnd that are tagged with an appropriate tag (e.g. amenity=animal_hitch) but are not shown there, you can ask the maintainers of OsmAnd to add the rendering in OsmAnd. I'm not familiar with OsmAnd because I don't map with my phone; but if you are an OsmAnd user, you can certainly do it yourself. I assume https://github.com/osmandapp/OsmAnd/issues is a good place for that.

If you don't know (and don't want to learn) how to retag your already mapped horse highline nodes from leisure=horse_riding to amenity=animal_hitch or something else, I can do it for you with your permission. But then the nodes will probably no longer be shown in OsmAnd until the rendering of the corresponding tag has been added there. Please let me know what you want to do.

Yes, good idea.

Hi,
you have retagged a building to an equestrian centre (leisure=horse_riding). This was a bad idea, imho, because
1. you removed the building=* tag from the building and it is no longer rendered as a building
2. the equestrian centre is larger than the building, the area should cover the entire site of the equestrian centre, like here: way/1293314373 (or map it as a node). Compare leisure=horse_riding

Hi,
way/1363704961 looks like a riding arena and not like a riding center (leisure=horse_riding). Riding arenas are better tagged with leisure=pitch + sport=equestrian, like way/311696114

What do you mean with "the map"? OsmAnd?