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.
The knockdown only happens on a successful hit.
Everything between the attack name and “Hit:” is general to the attack; everything after “Hit:” happens only if the attack hits.
This is detailed explicitly in the Introduction of the Monster Manual, where it explains how to read monsters' stats, so you don't have to rely on implicit reading at all. On page 11 under Actions,
Melee and Ranged Attacks (headings are on previous page) it says:
Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation.
Best Answer
You are mounted and prone
In this case, the originating effect is something where the rider now has the prone condition. When mounted and prone, the rider must make another DC 10 DEX save to stay in the saddle.
The rider remains prone (and still mounted) on a success, but falls to the ground and remains prone if failing the DC 10 DEX save.
Mounted and Prone - huh?
In this case, we've got a weird situation to imagine. The rider is prone and suffers all the effects of that condition. It's a simple mechanic with clear effects.
But what isn't clear is how do you narrate being prone while riding a horse.
That bits going to be opinion-based, but this DM would probably describe it as you're still mounted, but you're definitely not seated well and are trying to recover your balance. In this narration, it's being unbalanced that gives you the mechanical effects of being prone and still requires the necessary movement cost to undo the condition.
As long as you keep the mechanics of being prone, you can use whatever narration you'd like to describe what it looks like. But the condition itself remains until resolved.
Standing up while sitting down
The other weird piece to this is that prone is resolved by spending half your movement to 'stand up'. It is unclear whether 'standing up' when mounted means you actually need to dismount first or that 'standing up' is more figurative to just needing to expend movement to no longer be prone.
Because the rules here are unclear, it is going to be up to the DM to determine the resolution for the condition, but this DM would rule that it is the expenditure of movement that resolves it. The act of 'standing up' becomes "sitting up".