What Dolorous Edd says.
If I may ... having ridden horses and ponies since I was 18 months old (my uncle, my grandfather, my g-gf and my g-g- gf were all farriers, and I would have been one too if the WCF hadn't had such outdated apprenticeship rules)
The horse is the ultimate prey animal - it has minimal fight instinct, and overwhelming flight instinct, only somewhat 'bred out' by its millennia of domestication, and still needing intensive training and desensitisation.
A horse also has a very, very different field of vision - and 'way of seeing' - to ours, and an understanding of its visual ability helps us to understand many of the problems and difficulties encountered with horses in particular situations.
Something speedy and silent - a cyclist, let's not even talk about electric vehicles - coming up from behind (a predator! say all of a horse's instincts) will at least slightly startle even the calmest and most well-trained of horses. Yes, even the calm and confident ones who are totally accustomed to being accompanied by someone on a bike while someone else rides them. The calm and confident won't flee, but you might notice a lift of the head and the rider will feel a momentary tension in the horse's body, transmitted through to their own, and the steady rhythm of the horse's hoofbeats may be just slightly disrupted, with an off-beat in the walk, or even a momentary extra half-step as if one leg is thinking of moving to the next gait up.
However, horses are totally accustomed to, and confident about, being spoken to as people approach them from every direction. It is an unwritten rule among horsepeople that you never take a horse by surprise, always using your voice to notify the horse of your approach, your presence and of your position in its environment. The domesticated horse still needs your reassurance that you are not what its most basic instinct still tells it to fear - a predator.
Hence speaking - telling both horse and rider who you are and what you're doing, especially if you're coming up on a quiet road with little other traffic - is your best safety measure by far - for all concerned. 'Cyclist approaching, overtaking on your right, thanks have a nice ride' is better by far than ringing a bell as you need to communicate with the horse as well as the rider.
The really quiet road with little traffic is probably the most dangerous road on which to approach a horse to overtake it, as you are more likely to startle it with the cycle's silent approach than you are on a road which has a reasonable amount of traffic on it, when the horse itself will be more alert to everything around it.
When riders are doing that annoying 'riding two abreast', it's often because the horse on the inside is a youngster and/or inexperienced in traffic - and hence nervous - or maybe has had a frightening experience in traffic. Being alongside a calm, confident-in-traffic horse allows the other horse to learn, or to regain, confidence and many horses used for this sort of 'nannying' work will actively use their own body to help control and reassure the more nervous one, pushing it into the the safe side of the road or barring its forward progress and its urge to flee. Unless a horse is going to spend its entire life never venturing out into the world, it HAS to learn how to deal with traffic with some degree of aplomb; you can only do so much in the 'comfort' of the horse's own field or stableyard, and sooner or later it HAS to venture out onto the public roads, preferably in the company of a steady, 'traffic proof' horse. Naturally for those first 'baby steps' out on the road, you try to use a quiet, little-trafficed road - and quiet, little-trafficed roads are likely to be narrow with high hedges and bends ... you might even have arranged for a friend to drive up the road and overtake you at certain points, or have gone out deliberately at a specific time to eg avoid the milk tanker . If you have a young, nervous horse, it's no use at all hurrying it out of the way of any 'monster' approaching from behind, whether planned or unplanned, however much you want to allow the unexpected and impatient driver or bunch of cyclists to get past, as that only confirms in the horse's impressionable mind that getting away fast from the monster behind is the right thing to so. Neither would it be safe to fall into single file and thus remove the panic avoidance measure - the nanny horse - from the nervous one's side, even if that would appear to make it possible to overtake. The best and safest thing the rider or handler of the horse can do in that situation is to just carry on as if nothing at all of any consequence is happening , until they reach what the rider or handler judges as 'safety'. An experienced horse can often be relied upon to 'do the right thing' even with a nervous or inexperienced rider/handler; this works both ways though, and the nervous or inexperienced horse will often take a cue as to how it should 'behave' from the actions of its handler ...
