How did people wear these kinds of hoods without them flying off their shoulders when riding or running or even fighting? Were they secured with a leather strap over the shoulder or something?
Here's some other examples:
You wouldn't have to worry about a hood flying off for the same reason most people don't worry about their necklaces. To come off accidentally it would have to go straight up while maintaining its shape and avoiding the chin, nose and ears. A flexible fabric in a solid piece around the neck just isn't going to do that without using your hands. It might bounce around and get annoying, but it isn't going to just fly off.
They also may have been held down with string or buttons/toggles attached to the other clothing, but I don't know if that was actually done.
Depending on the style, these are either slip on, or have buttons running up the front (Russell here's version appears to be a mutant blend of the two). For button-up hoods, there's no chance of them coming off without being unbuttoned - the neck opening isn't large enough to go over the head (try this; attempt to take off a necktie without loosening it).
For the slipon ones, obviously the neck hole needs to be large enough to go over the head (so that you can slip them on...). However they tend not to have an enormous amount of space, so there's a lot of friction both between the sides of the head and the upper part of the hood and (to a lesser extent) the cape of the hood and your upper-body clothing.
As a side note, consider the chin; for the hood to come off, the lower front of the face opening has to come forward and then up to clear the chin - this isn't a movement that's likely to happen from running or riding.
