I appreciate the link. But my case is different in seemingly two important points.
1. the two Macintosh HD drives shown in the finder side bar both have the same name "Macintosh HD" and are identical in every respect; I have no "Macintosh HD-Data" showing in my finder.
2. I apparently have no problem with time machine. I received no message asking me to rename "Macintosh HD-Data". I suppose for the obvious reason that finder shows no such name. And I am able to do back up's on time machine without any problems and I am able to enter in on these backups and access everything without any problem.
Given this differences, I wonder if my case is simply something else or if it is not problematic enough to merit me having to spend another day reinstalling my mac os?
Thanks for the help.