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".