The video shows a very slow speed so there's no sudden acceleration here.
...why it would have misjudged so badly...
Tesla automation system (Autopilot/Enhanced Autopilot/FSD/AutoPark/Summon...) in Tesla's letter to the California Department of Motor Vehicles is defined as Level 2 which requires a competent driver to drive the car at all times including when the system is active.
It's called L2 because it is not competent enough to safely work on its own: It needs a human driver to intervene in such as this scenario.
As others stated, Tesla Auto Park relies on Sonars. Sonars work great as long as the environments fit its criteria. If you count your sonars, there are blind spots because there are missing ones on the side of your car. Your car's bumper alone has 3 sonars each facing rear or back. Your side of your car which is much longer than your bumper has only 2. One all the way to the front side of your bumper and one all the way toward the rear side of your bumper. There are no sonars near the side doors.
Tesla can fix this but it will cost more money and it will no longer be L2 that requires human supervision, and it'll be a higher level like 3 and above.
- Who will pay for the repair to the damage to the other car, us or Tesla?
Yours is not the first. I've never heard of a case that Tesla would pay for damages due to its L2 automation from the slow speed to high speed. That's the definition of L2: The system is not good to be L3 and above. Any car below L3 is the responsibility of the human driver to drive.
...How do we report to Tesla
You can go to the Tesla Contact page. The last time I checked, there is a phone number for EU customers but US customers can only chat by text on that page.
...because this sounds like some sort of bug that they should fix?
Yes. That is why it is an L2 system: An imperfect system that requires humans to fix their misbehavior in real-time.
It's cheaper to use car owner's labor to fix the bug in real-time with manual steering/braking/accelerating and keep it as an L2.
It is costlier for engineering's brain to fix the system so that it won't be L2 anymore but L3 and above.
...Given the only thing we use "Full Self Driving" for is the auto-park, and we've lost confidence in auto-park now, is there any way to get the "Full Self Driving" package cost refunded?...
I have not heard of any case that Tesla would refund the cost of its automation feature due to its being inferior to L3 capability and despite damages.
I don't think most courts in the world would be sympathetic to Tesla owners who didn't know what they bought because what they bought was described on the page with the price of the feature (Autopilot/Enhanced Autopilot/FSD) which says "require active driver supervision and do not make the vehicle autonomous".
However, the German court might be sympathetic because it banned the name "Autopilot" for misleading.
...Does anyone on this forum have any experience with this scenario?...
I would say there have been numerous accidents with Auto Park and Summon.
Owners should prepare their mind that "it will hit the car this time" and get the brake ready each time because if they don't, it might be too late to realize that it will hit this time actually.
I would say for those who have a good reaction time to correct the misbehaviors of an L2 car, it's a very good feature and well worth the money. However, for those who don't realize that L2 means it can still crash at any time including at a very slow speed of Auto Park, then maybe L2 is not for you.