You are knocked unconscious.
This one is obvious from the magic item description:
you can't be dismounted against your will if you're conscious
"Dismount" is a game term defined in the rules for mounting and dismounting.
Here are the things from the mount rules that the Saddle of the Cavalier prevents:
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. If you’re knocked prone while mounted, you must make the same saving throw.
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.
This is what the magic item description is referring to when it says "you can't be dismounted against your will".
Teleportation and getting swallowed are different from being dismounted.
Neither being teleported, nor being swallowed, are covered in the rules for mounting and dismounting, because dismounting is a specific thing defined in those rules. The Saddle of the Cavalier will not prevent teleportation or being swallowed, because those things are not the same thing as being dismounted.