Car horns seem to be calibrated to be heard by someone wrapped in a car, and are too loud for unprotected listening. Bicycle bells are closer to that range. No wonder the cyclist got freaked out.

The Ineos Grenadier 4x4 has a button specifically for this purpose (& because it's made by a cycling team & makes for good PR). I don't know how useable it really would be.

I don't think I'd ever use my horn to let a cyclist know I'm there. It's hard to do a friendly, quick double-toot consistently and, as you said, it's just too loud. It's more likely to cause an accident than just waiting longer to overtake. Horns are usually a rude thing, so they're scary to hear.

Similarly, on a bike, I don't really use the bell to go around pedestrians on shared paths. Pedestrians aren't usually in the mindset of being aware of what's behind them, so the bell just makes them jump. You also can't express the difference between a friendly warning and a request to move with just a bell. Again, it's easier and more consistent to slow down and take responsibility for moving around them.

(obviously, local cultures will vary)

As a cyclist, I find it hard. I tend to avoid using the bell for the reason you've stated (i.e. it tends to just make them jump), but equally I've found not using it gets responses along the lines of "Use your f**ing bell".

I don't have a bell, so I call out 'on your left', but you have to do it early enough that people have time to think about which side ks their left.

If you've got a freewheel, pedaling backwards can be useful; it makes a pretty distinctive, but not terribly loud noise.

I've found most everyone responds well to "on your left!" Many will wave and thank you, largely because most bicyclists provide no alert at all and instead just fly by (which is dangerous and rude).

Poor cyclist but the anecdote is hilarious