My understanding is:
Rider = rider is more experienced at the level than horse
Horse = horse is more experienced. All been there done that school horses would be in this category.
Open = for professional riders. If you take any money for training or lessons, you are by default a professional.
Amateur = this category is new, so I'm not really sure what it's for yet. Maybe people like me who rider and horse started at the same time?
No wonder you are confused--your definitions are wrong!
Typically the Rider division is for anyone new at the level. While technically you can have competed one level up (or more if it was forever ago), it's basically any rider who is at that level, regardless of the horse. So if you are a BN rider on an Olympic horse, you are in the Rider division. Typically (but not always) if you are a BN rider on a BN horse, you are in the Rider division as well.
The Horse division is the opposite. So if I were a high level pro bringing up a young horse, I would have a BN horse regardless of my level--so I would be in BN Horse (same rules--specific on how high a horse can have competed and when, but I would have to look them up).
The Open division is anybody. Not just pros. Open to any rider, any horse.
The Amateur is for someone who is a declared amateur and has filled out the appropriate paperwork for it. There are very specific and quite stringent rules on what can disqualify you from Ammy status, and you MUST meet the requirements AND fill out the amateur status when you register with the USEA.
So typically you will have an Open division and a Rider division to start out. If entries warrant you might have a Horse division or an Ammy division separately. You might have a Junior division--which can be horse, rider, or open (if they don't list it it's Open).
So my last show there were two divisions of Sr. Training Rider, an Open Training, and a Junior Training. There were no Horse divisions--they were all either in Open, Junior, or Rider. There were not enough inexperienced horses being ridden by high level riders to warrant a separate division from the Open or they just lumped them all in the Open division because it's easier if they need to move someone around.
When I enter a show I typically prefer Rider, Open, Horse divisions. If I qualify as an Ammy that year I prefer Rider, Ammy, Open, Horse.