Problem solved! With great assistance from Apple support, here's what led to the successful installation of 10.13.4:
After having downloaded the 10.13.4 installer from the App Store and proceeding with the Restart, the original problem again occurred. I depressed the power button until the computer shut down.
I then pressed the power button to restart the computer and immediately pressed and held down the command and R keys until I saw the Apple logo and progress bar. This put the computer in recovery mode. After a few obvious things to click, installation began. NOTE: this procedure did not erase the disk or any of my data, apps, preferences, etc. It was simply a matter of time waiting for the installation to complete.
When I got back to my desktop, I checked About This Mac (on the Apple menu), and lo and behold 10.13.4 was installed! I went to the App Store to see if anything was yet not updated. iTunes wasn't, so I updated it.
If 10.13.4 had not already been updated, I was told that once I got to the desktop, I was to go to the App Store and download the update and restart. In my case I didn't need to do that.
I certainly hope others have the same success! I'm feeling a lot of gratitude for Apple support. I also appreciate those of you who offered suggestions.