zehner please try without the case. It seems to use an internal adapter (board) for the HDMI ports, like the Argon one case, and those have a long history for causing issues like yours.
The first log shows that display detection on HDMI0 doesn't seem to work. The second one shows that it's fine on HDMI1. Most likely the hot-plug connection of HDMi0 is broken (could be the adapter board) - connecting the HDMI cable directly to the RPi4 will bypass that and very likely solve your issue.
so long,
Hias