SD-Card-Module doesnt work with external power supply
Okay, thank you all for your efforts.

Finally, after changing the PSU and connecting the Arduino to another usb-port without operating system, I found the solution.

The problem was the SD-Card itself:
I swapped the SD-card with a newer (maybe more expensive(?)) one and it just works now. No matter if you connect the Arduino with the PSU or over an usb connection.
My assumption is, that there could be a different clock speed of the sd-card module on different voltages, which causes the problem. The old sd-card can't handle the fast speed and runs into an error (or just rejects it) on a write attempt...

Anyway, thank you again for your efforts!