Thanks again!
I think I tried to access other admin pages as well, but the issue persisted. But yeah, if the module does not need to be loaded, then sure I should be able to access the admin's module refresh page, too.
Anyway, I have just tried the upgrade one more time and after the successful update page in the admin I tried to access the module refresh page (/module/?reset=1) by using the admin menu, and I got the error again, however, simply refreshing the page made the error go away and I was notified: "SystemUpdater: Detected core version change 3.0.266 → 3.0.267"
Could there be a way during the upgrade process when using the Upgrades module to automatically have the modules refresh, so that issues like this can be prevented in the firs place, perhaps?
Thanks for all the other troubleshooting tips too, I'll save them for later.