I further tried to use a "blank" Raspberry Pi Pico (only board, not used previously, no hardware attached).
If I run an example sketch from the TFT_eSPI library, and then read the User-Settings, the output is the same as above. (negative values for the Pins, pin numbers seem to be extended with a leading "1": MOSI pin is actually 11, not 111, MISO is 12,...)
I am not sure where the problem lies. If I flatten the waveshare sample code to one folder, I can successfully use it with the Arduino IDE and my setup. However, as soon as I try to use the same setup with @bodmer's library, I am not able to control the display except the backlight by using an analogwrite command.