Shame on me. :(
I forgot where I configured the E-Mail account used in my Drupal site. I guess I set up the E-Mail with the installation of Drupal, could that be possible? Does Drupal care about registering the E-Mail address and getting incoming mails?
I get all the mails at this address forwarded to my private E-Mail address, but I really don't know where I registerd the drupal address and where I told Drupal to forward mails to my private address.
In the Drupal dashboard I can't find settings for an E-Mail Account.