Hmm, that's a good question. I'm going to leave it up to you, the player, to decide which career path your photos and stories are most aligned with. I would say, if a horse and character are meeting with their coach, then the coach would get XP for the training career, the rider would get XP for their rider career, and the horse would get XP for either competition or leisure depending on the situation (we're thinking leisure horses will have some cross over into lessons and things when it's available).
If you're posting a somewhat ambiguous photo, like a horse being groomed, then I would say it's up to the context of the situation. For example, if he's being groomed for a show, then it could be applied toward his competitor career. The story you may write around any photos you post may help add clarity to the context too (and collect you some extra XP!)
Hope this helps!