Longtime horse rider and owner here, all over the world; I come from a family of farriers so always at the 'sharp end' of horses.
Use a ding ding bell (never a horn) by all means, but primarily , if you can, use your voice.
'Cyclist coming up behind!' Keep talking as you pass if you like, or if it looks (or you feel) nervous. Hello nice horsey, you're a lovely colour, I'm just a bike passing you, not a big bad wolf and I don't want you for my dinner ... or sing, or whistle. If the rider appears to be ignoring you, it might well be because she (it's usually a she!) doesn't want the horse to think there's anything going on out of the usual by responding in any way. It can be a fine line between reassuring the horse that everything's OK and not raising its level of alertness to a higher level by acknowledging that something's going on ...
Horses recognise the human voice and are reassured by it when used in a normal tone, not startled by it in the same way they are by sudden 'mechanical' sounds. They can then identify the swift silent thing coming up on them fast from behind as not a leopard or a wolf wanting its dinner, but one of those fairly benign things that bring it food and pester it when it'd rather be pootling about in a field ...
They will certainly see you - or rather, your movement - from a good distance away, even when you are behind them (they have a rather unusual, to us, field of view) but unless they can identify you for what you are, they may well be afraid of you. They are - or at least can be - afraid of all sorts of things; some the rider or handler will be aware of and some they will not. This is merely in the nature of the horse, given that they are the ultimate prey animal with little-to-no defence against predators other than their constant state of alertness, speed and stamina; time, patience and a degree of skill and understanding in the handler, together with some changes through millennia of domestication and controlled breeding, means that most of their issues can be overcome, or at least papered over.
If you're approaching from the front, it's often a good idea just to stop and wait while the horse passes you; again if it appears nervous just talking any old nonsense can help. It's the speed and silence of a bike - combined with its often-unexpected appearance just where the predator would pounce to bring its prey down - that can induce panic in a horse.
No, we horse riders would prefer by far not to ride on the roads and we would avoid them if we could - not just because of traffic, but because of the damage the constant concussion on hard surfaces does to our horse's feet and legs. But it's impossible to avoid them entirely even in the areas best-provided with bridleways. Even where they exist, bridleways often come to dead ends on busy major trunk roads, with no conceivable safe way of crossing. Many other bridleways are blocked and it's not possible - as it might be with a bike - to lift a horse over a fence, carry it up steps and over a bridge or tip it low to creep under branches. Bridleways which have been 'adapted' as all-user paths may have barriers which are difficult to negotiate for horses and have often been surfaced similar to a road. These all-user paths then end at some random point and may either feed out onto a cycle-cum-footway - often impassible for horses and certainly illegal - or, as we all know too well, peter out with a few worn markings in paint on a dual carriageway or similar.
