There's quite a few different scenarios here, let's deal with them one at a time:
1. Does a rider take falling damage?
It would depend on the height of your mount but most mounts are not 10 ft tall so you wouldn't take falling damage.
You quoted the relevant rule:
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. (SRD p.86)
Most mounts are classed as large creatures. A large creature could be up to 10ft by 10ft in size (SRD. p. 92). However a commonsense ruling is that large creatures are simply bigger than medium sized ones (5 ft by 5 ft) and don't take up the whole square. In any case it's unlikely that a rider would fall from the tallest point (it's head) of even an unusually tall 10 foot horse rather than it's back (which will always be a point lower than ten feet).
As it is unusual for a mount to be taller than 10ft no additional damage is normally taken.
2. If the mount is knocked prone is the rider also knocked prone?
If you don't take any damage from the fall, under the general rule a PC wouldn't be prone.
The creature lands prone, unless it avoids taking damage from the fall. (SRD p. 86)
However this is a case of specific beats general:
If your mount is knocked prone, you can use your reaction to dismount it as it falls and land on your feet. Otherwise, you are dismounted and fall prone
in a space within 5 feet it. (SRD p. 99)
If you choose not to use your reaction, or cannot use it you are knocked prone despite taking no falling damage.
3. If the mount is forcibly moved against the rider's will are they knocked prone?
If an effect moves your mount against its will while you’re on it, you must succeed on a DC 10 Dexterity saving throw or fall off the mount, landing prone in a space within 5 feet of it. (SRD p. 99)
In this instance the rider can make a saving throw to stay mounted, but once again specific beats general, so if they fail the saving throw they fall prone even if the don't take any fall damage.
4. What if the rider is knocked prone while on top of the mount?
Despite being prone they have the option to not actually fall off the mount:
If you’re knocked prone while mounted, you must make the same saving throw [as previous quote]. (SRD p. 99)
If they pass they're 'prone' on the mount, if they fail then they're prone on the floor.
Being prone on the mount may sound odd but there's nothing in the prone condition that states it can only occur on the floor.
5. What about mounts taller than 10ft?
The same rules as above apply - only this time you will also take falling damage, which would knock you prone either way.
6. Of course, all of this is worse if it's a flying mount...
The rules as above apply - you just risk taking a lot more falling damage and either way you'll be prone.
It'll be more the height at which you're flying that's the issue than the size of the mount you've fallen off of (though both are relevant). If you presume that the player is sitting on top of their mount (rather than hanging on underneath it) then the fall will begin from the top of the mount and not below it - so falling distance would equal the height of the mount above the ground plus the from which you fell on the mount itself.