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.
I use this, with DC = 10 + the spell's level
And I have used it since before I read the rules in XGtE with absolutely no problems at all and my group really appreciated it.
This is also exactly what passive checks are for. If you notice something your brain immediately makes the link to your knowledge and you just know, you don't need to stop and concentrate to know something, you either know it, or you don't.
Notes
This makes Counterspell more powerful than it seems to be intended in XGtE, but I think that rule sucks and having a spell countered, or countering a spell is more fun than wasting a counterspell. But then as a GM I am on the players side and like them being informed about anything and everything.
If using DC10 as the base drop the advantage rule, because that boils down to +5 and means you start on a base of 15 passively so pretty much recognise everything.
I did try DC = 15 + the spells level, and this has the effect of making non-arcane spells more difficult to counter. I didn't like this personally, but it can be a flavourful touch. The DC15 is essentially taking into account that you have advantage on your own type of spells. I think here it is a group preference thing.
TL/DR
I find that more information in the hands of the PCs is better than less information, and any clever use of a passive skill such as this is very much within my style of DMing, but each to their own and this is steering away from RAW.
Best Answer
There are a few implications but most of them are just aggravating already existing factors.
1. Positioning will become even more relevant.
Positioning is already relevant but when a fall causes even more damage some risky places become deadly places. The fall from the top of a 40ft. rampart that caused 4d6 now causes 10d6 damage, what could hurt your character after a shove now can kill him. Every edge from a high place becomes even more important to not get close to it or more rewarding to use offensively.
2. Feather Fall just became more necessary.
Feather Fall already is one of those spells that you always want to have because you never know. A 6th level wizard that was just worried to be throw down 60ft after flying now should be terrified of the idea; he might survive 6d6 but hardly would upon receiving 20d6 damage.
3. People that can avoid falling damage got comparatively more mobility.
You're chasing a 4th level bard and he jumps of a 50ft building, cast Feather Fall and safely reaches the ground. A Fighter or Barbarian might be able shrug 5d6 and keep the chase but even a monk with Slow Fall would think twice in jumping down when the damage is 15d6, ignoring 20 damage, the average of 15d6 still would result in 32,5 damage. The same monk could jump into an existing ledge 10ft under and then to the ground and probably get off with 15 damage, but his companions can't easily follow unless they slowly climb down or have the similar resources.
4. Athletics and Acrobatics became even better skills.
They still aren't Perception, but one of them reduce falling damage and the other helps you to climb up or down and not fall. When falling damage just got lethal, not receiving it becomes more important than before, raising the value of those skills.
5. Spells that can throw you instead of direct damage got a lot more lethal.
Telekinesis, Reverse Gravity and Bigby's Hand can kill a lot more effectively to the point of being almost broken since they can continuously cause 20d6 with a single spell. Beware of those.
6. Knocking flying creatures prone becomes more lethal.
Any ability that knocks someone prone becomes far more lethal since non hovering flyers will fall as a result and take much more damage. (@OganM's point).