Do both HD 3850 have same exact Video BIOS ?
Some (very few) boards/bios + vid cards/video bios do not proper reset on a cold start or warm reboot. Have a newer PCIe that will make saw tooth patterns on rare occasion on a cold start but never on a reboot on one specific motherboard.
Your issue seems to be opposite but am thinking they are related in same way.

I don't know if I understood you correctly. But I got a little hope that I can fix it...

No the other card works well on both ports... The one with the error is the sapphire model - but there is an aftermarket cooler installed zalman vf700 or so... The one that works well is the powercolor model.

I don't understand how the bios can cause these problems. And how can it be checked? Why does it occur every first time I reboot... But not the second time...? Like something isn't initialized that's needed... Just a guess
It would be really nice to get it fully working again... 🙈

Oh my God... Maybe the card is ok completely... I just exchanged the DVI cable and after about 20 reboots the error doesn't occur. That would be great news... Will test it in depth this evening.. I have high hopes

Ok .. after many tests... 5 different cables... Both ports... Shortly suspecting the Monitor to be broken... It seems clear... Something's definitely wrong with the first port if the graphics card. But I really don't know why this error comes every second soft reboot. This is ..... I don't know...
Should I try flashing the latest original bios?

If you are handy with the soldering iron? Try re-flowing all of the pins on the connector with some leaded solder.

Repeated connections over the years often weaken the joints.

texterted wrote on 2021-02-07, 03:15:

If you are handy with the soldering iron? Try re-flowing all of the pins on the connector with some leaded solder.

Repeated connections over the years often weaken the joints.

Unfortunately I am not...
Or better: I don't know if I am - I never tried.

But I am still wondering why this occurs so regularly

Well, I wouldn't learn using a piece of hardware that you care about!

Lazar81 wrote on 2021-02-06, 22:22:

Ok .. after many tests... 5 different cables... Both ports... Shortly suspecting the Monitor to be broken... It seems clear... Something's definitely wrong with the first port if the graphics card. But I really don't know why this error comes every second soft reboot. This is ..... I don't know...
Should I try flashing the latest original bios?

Hello again... I went on trying this for days and days... It took a while for the easiest idea to come to my mind. I just tried another monitor. After days of excessive rebooting the error doesn't occur... Back to the former monitor brought back the error immediately.