+1 to tolerance.
Horses were on the road before even cyclists were, indeed many of the lanes we ride in preference were designed for horses originally. They have a special status in the highway code, recognising the fact that they are live animals and not vehicles and as such special consideration is essential. Road users who ignore this do so at their peril since it is not just the horse and rider they may be putting at risk but themselves too.
If getting off my bike and speaking softly so a spooked horse can see I am a human being helps everyone get where they are going safely then I think it's the least I can do.
During my training I spent a bit of time working at the mounted police unit in Long Ashton, Bristol. Of course crowd control is a different matter altogether than the normal roads, but they have an entire garage packed with props for desensitisation training for the horses. It is incredibly hard work running through every possible sight, sound, object and type of movement and this has to be regularly repeated and worked on to keep these horses as bomb proof as possible when out and about.
Every day they would build a special assault course the horses had to be ridden through, involving flags waving, someone wrapped in a space blanket, walking over bubble wrap and other fabrics. They had bikes, prams and various items too. Each horse had it's own 'weakness' which was sometimes the most unexpected thing. Quite a few of them had a problem with shiny objects (which I believe can be an issue with bikes).
There are probably many horses that are OK with the most common traffic, people, dogs and street noises but rarely see a bike and so the rider won't actually know how their horse is going to react. Don't forget that we often use the lanes because we hope not to see much traffic. The horse riders have much the same idea as us only we are part of the traffic that they were hoping not to meet.