There's no such thing as a "daily updater", and it never "update the updater".
There's a daily CHECK to see if there IS an update available but, of course, it's not as if we publish updates every day so, if there are no new updates available, it won't say anything. And, even if there are, it will ask you to exit the sim and run the FSDT Live Update.
Years ago, when the updater only updated GSX, it was possible to apply SOME kind of minor updates without restarting the simulator, but now the Live Update updates everything including executable and sceneries, it always require to run separately when the simulator is closed. And in that case yes, it WILL "update the updater", since the Couatl_updater.exe won't do anything else other than downloading and starting the Couatl_updater2.exe, which is what does the actual update job.
