Problem solved. The plugin that interfered with the view count was Picture Perfect. It is listed in Extend as a developmental plugin, with a warning to not install on a live forum. After removing it, I looked at threads that had images in it, and they showed up as attached thumbnail images. The threads are viewable.
For the benefit of others who are trying to fix a problem caused by a plugin, here's what I did.
From File Manager in cPanel, I looked at the files in inc/plugins, then sort by date.
Remembering from previous investigation that the last date of thread or post that showed more than 0 views was 9/16 at 12:54, I observed the Picture Perfect plugin was installed at 12:59 on that date. Ah ha!
However, not everyone has such a clue, so I did a full process of removing plugins manually. This was required since pluginuploader was installed and only permits superadmin access in MyBB.
First, I created a folder called holdthem in the plugins folder, then used cPanel's FileManager to move all *.php files to holdthem folder. This disables all plugins.
Second, I went to the forum and observed that clicking on a thread in a forum, then clicking back and refreshing the page incremented the view count showed it was working properly as expected.
Third, one-by-one I moved a plugin php file from holdthem folder back to the plugin folder, and verified forum still working as in step two.
Finally, when I moved the Picture Perfect file back to plugins and the view count did not increment, I verified the faulty plugin.
It is interesting that Picture Perfect on a fresh installation of 1.8.27 does not exhibit the view count problem. So there is an interaction somewhere in the system perhaps in conjunction with another plugin that could be the root cause. This would require much deeper troubleshooting that is beyond me at this time.
Thank you Mr. Friday for the trust you've shown me in granting access to your cPanel. I did not save the password, and I have deleted the PM which contained it.
In the process of enabling the plugins one-by-one, I confirmed that it is the MyAdvertisements plugin that is causing the double back-click necessary to go to the previous page. Again, not tested with a different browser, but evident in Firefox.