Mysterious transmission problem

Hi everyone, I have enjoyed my 2004 Prius for years and have never had any problems with it. Last month a mysterious problem suddenly arose that I cannot solve. My hope is that someone here knows this Prius so well that s/he could help you identify where to look.

# Event 1: All this happened on an evening within about two hours.
After driving 1.5 km and stopping in front of the traffic light, the car refused to drive further. The car displayed a red triangle on the dashboard. After a lot of trying it was possible to ride a bit (into the roadside). [I was not present so don't have many details]
Later it was taken away by the roadside assistence AAA (ANWB in the Netherlands) on a trailer. It was thus moved a little in N (neutral).

Later this happened (I don't know if it was still powered-on or if it was first switched off again): it again gave a red triangle warning in the dashboard plus the following message on the computer screen: "batteries cannot charge in N, must put in P." If I put it in D or R, it shot back to N (and the error message thus remained). Then was able to turn off the car.
The AAA has tried to read error codes with two devices and searched their knowledge base, but couldn't find this problem.
Reading out error codes only resulted in a low emission value (catalyst). This was probably an old code from a while back when the oil level was low and therefore, if you made a sharp turn, the engine gave a warning (red triangle, and orange engine light). After topping up oil, this was completely remedied. This happened a week before. (This code was not seen again afterwards))

After that, the problem with the car could not be reproduced regardless of trying.

Later only the ignition was switched on to be able to charge a telephone (one on / off button, no brake pressed). Moments later, with the brake pressed, the on / off button was pressed again (this should power on the car completely). It booted further, but did not display any speed (e.g. 0 km / h) nor display the message "Ready". That is not correct and therefore it cannot be driven around. When pressing the on / off button again, the car refused to go out! It was kinda stuck.
Reading error codes resulted in a new "immobalizer" (star blocking) error message. There was also something like a "key code" report, possibly because another Toyota key was nearby. (These codes were not seen before and after.)
Only with a very long press of the power button did the car eventually go out.

After that the problem with the car could not be reproduced.

Later, by trying to switch a lot with the stick, it was possible to produce the same. Then came the red triangle message "in N it does not charge, put it in P". Then it was possible to put it in a different gear according to the AAA guy (but maybe he meant to put him in P). The message disappeared as a result.
This yielded no relevant error codes.

Then we drove home with no problem.

# Event 2: This happened the next time driving (a week later).
Drove for half an hour without a problem. After shopping, we tried starting the Prius again. When the gear was set to "D" this resulted:
The red triangle appears in the dashboard, a continuous warning sound is heard, and this text appears on the on-board computer: "Attention. Press P when parking. The batteries are not charged when the transmission is in Neutral N".
This warning was audible in "N", "D", or "B".
The warning disappeared in "P" or "R"
Moreover, it was noticeable that it was possible to drive it in "D", "B", or "R". But with all warnings (and warning sounds) that doesn't feel right(!)
Also noticeable is the 'door open' warning in the dashboard. Does the car think that someone opens the door while driving and therefore that you are not parking correctly? I cannot remember whether the parking brake was really on at this time.
There was no getting away from the warning, no matter what was tried: switching gears, driving, turning the car off and on.
Then, after the car was off for two minutes, restarted: then the warning suddenly had disappeared.
[Could this be because the car, if you switch it off, is still active for a while (you can operate the windows electrically for another half minute after switching off the car) and only then do the systems really go out?]

# My summary of the problem
-Problem seems to arise in my analysis somewhere around the switch (self switch with stick, auto switch)
- Problem seems to arise when standing still.
- Sometimes it is possible to switch to other gears and sometimes not (shoot back in N).
- Accordingly, it is and is not possible to still use it (?)
- It seems to lie outside the on-board computer's analysis capability because there are no recurring error codes.
- Sometimes it gets stuck (can't switch on / off normally with power), sometimes not.
- You can get rid of it by: full (10sec) reset, by switching off and waiting a long time before you put it on again and .... luck (?)

Is it just a software error? Is it a sensor? Is it a thread that is loose? Or a sub-computer that is broken?

Notable is this: in the pas I have had some water leakage in the rear trunk. Perhaps something happened there?

Who-o-who knows how to help me further? Many thanks!