How do I change Music Library location on Mac mini with macOS 14.6.1?

See Managing your Mac media libraries - Apple Community and Make a split library portable - Apple Community for some background here. If you've moved or copied the entire ~/Music/Music folder to the external drive then hold down option when launching Music to select the .../Music/Music Library.musiclibrary package that holds your library. Changing the media folder location in preferences doesn't change where Music expects to find content that has already been added to the library. Consolidating the existing content to a path that you've already copied media to will generate unwanted duplicates, and splitting the library by keeping the database on the system drive while the media is copied to an external will complicate your next migration of the library to a new machine.

tt2

Go to Music > Settiings > Files

Change the location to the external drive

Then file > library > organize library > consolidate

Your music will move to the new location