There are several things you need to clarified to understand your current situation. The iTunes library file is a database that keeps track of you iTunes media. It defines your iTunes library. It is usually called iTunes Library.itl. The iTunes folder (in your startup disk user account Music folder) is the default location for iTunes Library.itl file.
Then, there is your iTunes Media folder. By default, this folder contains all of your actual iTunes media files, organized by iTunes into subfolders by media type, artist, and album. The default location for the iTunes Media folder is also in your iTunes folder (in your startup disk user account Music folder).
One common option to move your iTunes data off of the startup disk is to put your iTunes Media folder on an external drive (or network storage) while keeping the iTunes Library.itl file on your startup disk at the default location. To do this without iTunes losing track of your media files (where they are located), put an empty iTunes Media folder at the target location. Change the location of the designated iTunes Media folder in iTunes preferences Advanced pane to point to the new iTunes Media folder. Then give the "consolidate library" command to have iTunes copy your media files and organize them into the new iTunes Media folder location. This command is at menu bar -> File -> Library -> Organize Library -> Consolidate files (checkbox).
NOTE: In iTunes preferences Advanced pane, the TWO checkboxes for Keep iTunes Media folder organized AND Copy files to iTunes Media folder when adding to library should be checked (unless you have a specific reason for not doing so).
The OTHER common option is to manually copy (using Finder) your entire iTunes folder (with the iTunes Library.itl file and iTunes Media folder) to the external drive or network storage. Then, use the option to open a different iTunes library
Open a different iTunes Library file or create a new one - Apple Support
to select the iTunes Library.itl file at the new location.
The key difference is the location of the iTunes Library.itl file. The iTunes Library.itl is frequently accessed and updated by iTunes. If it is located on network storage (with much slower access speed compared to a directly connected drive), it can adversely affect the performance of iTunes. Therefore, the first option described above may be preferable, to keep the iTunes Library.itl file on the internal drive while storing the media files on network storage.
In your situation, which setup for iTunes is being used?