What Happens if the Rider undergoes forced movement?
PHB pg 198 also has a partially relevant sentence:
If you’re knocked prone while mounted, you must make the same saving throw.
While none of your above examples involve the 'Prone' status affliction, we can reason that under normal circumstances, if something were about to try and force move you (effectively off your mount), then you would be subject to the same saving throw to remain mounted as outlined above as you would if the mount were also being forcefully moved. (DC 10 Dex).
What about the military saddle?!
If you add in the military saddle, then you would have advantage on the DC 10 Dexterity saving throw to remain mounted.
This is an example of 'Specific beats general' rule of thumb. Were the general rule is found in spells like Thunderwave, that do not allow you save, they just work, and push you up to 15 feet; and the specific rule is what happens when you are mounted and you (or your mount) are forcefully moved while using a military saddle.
What about being grappled in your military saddle?
The grapple rules are on PHB pg 195:
"...you try to seize the target by making a grapple check, a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you succeed, you subject the target to the grappled condition..."
Grappled: Your speed becomes 0...
Where as the other examples are doing something to impact you; a shove, a shockwave, a hard attack, the person grappling you is physically picking you up and moving you against your will (even though your speed is 0). So I would personally rule that you gain advantage on the check (because of the saddle) to avoid the grapple condition but once you fail and become grappled, you are at the mercy of your captor. You can continue to save and break free of the grapple, but you get no more advantage, and can do nothing to control being forcefully moved out of the saddle as you are grappled and no longer able to hold on to anything.
Application of the above example: Let's say a Kraken tried to grappled you (shame on you for riding your horse on the ship deck!), you could try and hold on to the saddle and fight away the tentacles, but if they overcame your resistance and wrapped themselves around you; you would be effectively helpless to do anything that would gain you advantage once it tried to pull you out of the saddle and see how tasty you are. You struggle as you are lifted out of your mount and over the water, drawing closer to the creatures maw, but you break free on your own just before you become Kraken food.
Best Answer
There's nothing in the rules that mentions any sharing of reflex saves between mount and rider. Each rolls a separate reflex save. Unless it's stated specifically, a mount and rider do not share any abilities such as Evasion with each other, either.
To address the issue of realism clashing with the game rules, I don't think this is out of bounds with realism, either. While a reflex save is described as "moving out of the way", the act of rolling a reflex save doesn't grant any actual movement. Any movement that's used to describe the reflex save would be within your own space.
I can think of a lot of descriptive reasons why a mount that occupies a 10 foot square (such as a horse) would have a different save result as the rider, despite the mount moving the rider with it. Perhaps while the horse was moving to avoid a fireball, it left the rider more exposed? Perhaps part of the rider's successful save was using the horse for cover?
The relationship between the mount and rider also only applies to reflex saves if something specifically states it. If your character has a special mount that has an ability that allows its rider to use its reflex save, or its evasion ability, then it would apply.