What happens if a Cavalier or their mount are knocked prone?

The rules for mounted combat (PHB 198) give direction in the following cases:

  • Mount moved against its will: rider makes Dex save vs falling off and landing prone
  • Rider knocked prone: rider makes Dex save vs falling off and landing prone
  • Mount knocked prone: rider uses reaction to dismount standing, or else is dismounted and falls prone

The Cavalier’s "Born to the Saddle" feature (Xanathar 30) states "If you fall off your mount…, you can land on your feet if you’re not incapacitated." How do these features interact? Specifically:

  • Mount moved against its will: if rider fails save, they land standing up instead of prone. This is clear-cut; this is not the question.
  • Rider knocked prone: Normally if the rider succeeded the save to stay on the mount, they would be mounted and prone, per this question. If a Cavalier is knocked prone while mounted, do they end up either "mounted and prone" or "unmounted and not-prone"? It seems weird that a creature could be knocked prone but then just decide not to be; is giving up their mounted status the tradeoff?
  • Mount knocked prone: unless they spend their reaction, PHB says the rider "[is] dismounted and fall[s] prone." Does this count as falling off the mount for the purposes of "Born to the Saddle"? And therefore, can Cavaliers dismount for free (instead of using their reaction) if their mount is knocked prone?