Okay re-reading that I suspect that the mapped network drive isn't available when you start MC on your workstation, so MC defaults back to a local drive.
If you believe that the HTPC is up and running, and therefore the mapped drive should be available, you will need to do some network diagnosis. You could just open Windows Explorer and check that the mapped drive is available before opening MC, even if just a few times to check that it is, and if the problem still occurs when it is.
BTW, when you say;
Also should remame the library to 23?
I assume you mean the sub-directory where library backups are stored by default. Something like "C:\Users\[UserID]\Documents\JRiver\Media Center 21\Library Backups" for MC21?
You should definitely not rename that sub-directory to "C:\Users\rgi\Documents\JRiver\Media Center 23\Library Backups", but if you look in "C:\Users\rgi\Documents\JRiver\" you see that there is already a "Media Center 23" sub-directory, and there should already be a "Library Backups" sub-directory of that. Just change the setting in MC File Locations to "C:\Users\rgi\Documents\JRiver\Media Center 23\Library Backups", and future automatic backups will go in there.
If you actually meant the name of the library itself as shown in MC, then yes, go ahead and rename it as you like. I don't put the version number in my library names as I like to keep them short but meaningful. I name them something like;
"[PC Name] [Local/Server] Library"
So I can see immediately if they are local or server-based libraries, and I add in stuff like "Direct IP" when it uses a direct connection, for example. After all, all the libraries in MC23 are MC23 libraries. I don't need that in the name. I separate the versions of MC library backups by the sub-directory structure above, so I can always get back to an earlier version if I need to.