OP: Thank you all for the above suggestions. I did find a fix.

Visio had about 70 or 75 registry entries. When I uninstalled the program, there were still roughly 30 or 35 entries related to Visio.

I found a program called Registry Finder (www.registry-finder.com), which is an improved substitute for RegEdit. With Registry Finder (in Admin mode) I was able to delete all the remaining Visio registry entries. Then I reinstalled Visio 2013. It installed fine and started fine in normal mode.

There had been a particular document that was causing problems, before the big problem cropped up. It seems to be acting fine now as well.