Hopefully, I can clarify. And please believe me, I understand why this is confusing. And YES, I believe your issue is resolved.
First, please note that some types of updates do NOT require a restart to be installed. With your previous "Automatically download and install" setting, the update would be installed and you might not even know about it unless you know where to look. (Apparently, you now know where to check for updates but even then, some of them are hard to find.) Occasionally, those same updates will require Windows to reconfigure itself when the computer starts the next day.. That explains why you might see the "Do not shut down the computer" message and possibly a percentage display of the configuration process.
Second, please note that to the computer, a shutdown, then a restart the next day is the same as a 'reboot'. No secondary "reboot" is needed. As you've indicated, when you visit the Windows Updates site, then download and install the updates that have been detected during the update scan, you are asked to restart the computer, which you normally do immediately. (I do as well) During the reboot of the computer, you should notice the computer tells you not to shut down the computer and you'll also see a percentage display which generally proceeds to about 33%, then the computer shuts down and it restarts. Upon the restart, you'll once again see the warning against shutting down the computer, and the percentage display appears again, this time proceeding to 100%, after which you'll see the login screen. Such is normal when using the Windows Updates site and it's why I tend to use it exclusively, instead of automatic updates.
On the other hand, with the "automatically download and install" setting, the updates are downloaded automatically but are usually installed just prior to when the computer is shut down. The next morning when you start the computer, the only thing you will see is the "do not shut down the computer" warning and possibly the percentage display of the configuration process. In all of these cases, you won't see the updates listed as successful until after the configuration process is finished. This should also explain why you see the "Do no shut down the computer" after starting up in the morning, and also why you might not have seen the installed updates listed.
Hope this helps. Or maybe I confused you more......
Grif