$customer_logout option must not be empty, this is what the message says. Someone or something must have changed its value.
I problably won’t send any WebHook for this purpose, or at least not too soon.
I just hope there won’t be any hardoced checks in the future that could create a bug. This seems highly unlikely though, due to the extended and consistend use of current_user_can().
Therefore I’m going to use “author” and this ticket is resolved. Thanks!