@ Istanbulu... Don't know if you've figured out your problem or not. If not, this may help:
First, you don't state (but I figure you know) that you can't simply move music files out of your MacMini to your external hard drive. If you do, iTunes won't "know" where your music files are; the pointers in iTunes will still look for the music files on your Mini, but the files won't be there anymore. In other words, the music files "in" iTunes are Managed while the music files on a different drive are Referenced.
Because of the way the iTunes database and content are handled (no, they're not the same thing) it's a little different to move music files that are already on a drive other than your main hard drive -- but that doesn't sound like your situation.
What you're trying to do is turn your previously Managed files into Referenced files. You can do this within iTunes, but not with Finder.
The answer (as you suggest) starts with iTunes Preferences - Advanced from the iTunes menu. Change the iTunes Media folder path to the (new) location you want. Then, from the menu bar, select File, Library, Organize Library and check the Consolidate Library box.
You are now getting to the "point of no return". To proceed, click Continue and your files will be copied to the new location AND the iTunes pointers will be changed. Not moved, but copied, and iTunes will now look for them according to the new path. Also, this process will only change files that were in iTunes to begin with; any other content files are still in their same location, and you'll have to use Add to Library later.
When it's done, make sure your content files really are in the new location. (You haven't moved the iTunes Library database, just the content.) If the files have been copied successfully you can delete the originals from your 'puter and reclaim the hard disc space.
This should take care of it for you.
First, you don't state (but I figure you know) that you can't simply move music files out of your MacMini to your external hard drive. If you do, iTunes won't "know" where your music files are; the pointers in iTunes will still look for the music files on your Mini, but the files won't be there anymore. In other words, the music files "in" iTunes are Managed while the music files on a different drive are Referenced.
Because of the way the iTunes database and content are handled (no, they're not the same thing) it's a little different to move music files that are already on a drive other than your main hard drive -- but that doesn't sound like your situation.
What you're trying to do is turn your previously Managed files into Referenced files. You can do this within iTunes, but not with Finder.
The answer (as you suggest) starts with iTunes Preferences - Advanced from the iTunes menu. Change the iTunes Media folder path to the (new) location you want. Then, from the menu bar, select File, Library, Organize Library and check the Consolidate Library box.
You are now getting to the "point of no return". To proceed, click Continue and your files will be copied to the new location AND the iTunes pointers will be changed. Not moved, but copied, and iTunes will now look for them according to the new path. Also, this process will only change files that were in iTunes to begin with; any other content files are still in their same location, and you'll have to use Add to Library later.
When it's done, make sure your content files really are in the new location. (You haven't moved the iTunes Library database, just the content.) If the files have been copied successfully you can delete the originals from your 'puter and reclaim the hard disc space.
This should take care of it for you.