I think the general rule is (or should be if one has any common sense) is that the thing with the most momentum (mass x velocity) is the thing that needs to look out for, and give way to everything else, because it's the thing that can do the most damage to the other things, with the least damage to itself.
So, on a road, it is beholden on the motorised traffic to be extra vigilant about cyclists - I believe that if there is any doubt motorised vehicles should give way to cyclists - if a car clips a bike, a worse case is that the car needs a bit of filler and some paint. Worse case for the cyclist is rather more serious.
On a shared use path - as I keep reminding my kid - the cyclist is the heavy, fast thing, so the cyclist is the one who has to take extra care and stop if in doubt. We should ride as we would expect motorists to drive. I see people using shared paths and bridleways in this way less often that I would like.
Horses are a different case - they're a ton of skittish meat on sticks, with a brain the size of walnut and an intestine with the volume of a shed. Everyone should keep well away from them.