So the above configuration you picture,
To my understanding,
You can check either off or Both.
The first option,
Save Audit Records to Gateway Database
This is the default audits so it updates the mysql audit_main and audit_detail basically the ssg audit_* tables. Which tend to fill up mysql if you are not frequently cleaning audits or have not increased your mysql partition allocation.
The second option
Output audit Records via audit sink policy
Creates an internal Audit sink (and audit lookup) which are configured and typically used as a remote logger for audits. The default behavior of the Audit sink policy is if it fails to also write the Audits back to the internal database (same as option 1) unless configured otherwise there are cluster properties to not fallback.
In the event you have both checked off both sources are being written to. In the case you have Output audit records only the custom audit sink is being used unless the service fails and then it falls back to the internal database.
But on the highlighted option only you should be writing to audit_* tables.
Hope that helps.