How can the correction made to this album artist listing on PC #1 be migrated to PC #2?
An interesting question, and as I have never been a serious WMP user, other than as a simple player, I can only make some educated guesses. These follow.
MC can't read the WMP Database. MC has its own database, and it is stored on the local PC. Even if you use a MC Server and Client, the Client has a synchronised copy of the server database on the local hard drive. It is possible to configure MC other ways, such as putting the MC Library (Database) on a mapped network drive or NAS, but then if multiple Clients are using the one set of Library files at the same time, you
could will break the Library, due to file contention.
Therefore, your NAS is probably sharing the media files that it has via its internal DLNA Server. It could be using the WMP Database as the source of the DLNA information (I don't know), but it would still be sharing them to MC via DLNA. That means that each of your two PCs have a local copy of a MC Library, and they are using that to know what media files are available, and to store all tag changes made on either PC. So you have two separate, and not linked MC Libraries. (The MC Server functionality is the link between multiple copies of a MC Library.)
I'm pretty sure that tag changes made to files that are served from a DLNA server are not propagated back to the media files themselves (not positive), and probably not to the WMP Database if the NAS is using that.
If tag changes made in MC were written back to the media files via DLNA,
or if you are not using DLNA or the WMP Database in any way with MC (quite possible) but are in fact viewing the media files directly on the NAS from MC via either a mapped network drive to the NAS, or via a network address, and then each instance of MC on each PC could write tags back to the media files, and those would then be picked up by the second PC when it detects the changes to tags in the media file. That would be essentially doing some of what the Client/Server setup does, but with possible file contention issues, if both PCs try to write tags to the same file at the same time. You would need to at least have the following two settings;
1. Options > File Location > Cover Art > Also store image in the file's tag (This wouldn't work for videos, which can't store Cover Art in tags, or for any file type that can't store the tag data.
2. Options > General > Import & Tagging > Update tags when file info changes.
Anyway, I think what you have set up now is my Option 2 in my original reply above.
Basically, WMP and its Database are directly equivalent to MC and the MC Library. They are competitive products. WMP doesn't read the MC Library, and MC doesn't read the WMP Database (I'm pretty sure.)
Also, usually the WMP Database is stored locally on a PC, and not on a NAS, I think. Maybe some NAS' can store the WMP Database and share it to multiple WMP instances on multiple PCs. That level of integration with WMP may be supplied by some NAS manufacturers.
Tell me, with your current WMP setup, do tag changes from PC1 propagate to PC2?
NB: Again, I disclaim any knowledge of WMP and its Database, or detailed knowledge of DLNA, so these are just my thoughts.
PS: BTW, the correct answer to your question;
How can the correction made to this album artist listing on PC #1 be migrated to PC #2?
Is to use MC in a Client/Server configuration, which is why you need a MC Server. (As distinct from a separate hardware server of course. It could just be one of your two PCs, as per my Option 3 above. The MC Server is a software server, and may or may not require a separate hardware server. My MC Server is my HTPC.)