I was trying to understand the way your exception handling is working. I set everything to log my exception, but was never logged. I start debugging the core code and I found in the \concrete5.6.1.2\concrete\startup\exceptions.php line 8 that the logs table name was misspelled. In the code it's Logs but the table name is logs. So I changed the exception.php file and now the exception is logged.

Hope this will help !