It looks like I have figured out how to move iTunes Media (music, podcasts, movies, TV shows, etc.) from one external hard drive to another. Or, for that matter, from an internal drive to an external drive. Also note that in this case, all the iTunes program library files will remain on the current drive, which originally is the internal hard drive.
1) Do not copy the files manually. Do not use Carbon Copy Cloner or any other software. iTunes will copy the files.
2) On the drive to be selected, create a folder and name it 'iTunes', 'iTunesMedia' or similar.
3) Go to iTunes/Preferences/Advanced and select the location for the new media folder created in step #2. The location will appear as /Volumes/[Name of drive]/[name of folder]

3) Be certain to also check 'Keep iTunes Media folder organized' and 'Copy files to iTunes Media Folder when adding to library'.
4) Click 'OK'. At this point, iTunes will update the iTunes library information and now points to that new external drive. But no existing media will be copied to that new external drive.

Stop here if you want to keep existing Media in the former drive, and only copy new Media to the newly selected hard drive. This allows people to split their media between hard drives.
However, if one wants to consolidate all Media into the newly selected file, continue with these steps:
5) Go to Files/Library/OrganizeLibrary. Select 'Consolidate files'. If desired, also select 'Reorganize files in the folder iTunes'. This last option separates the Media into Music, Podcasts, Movies, etc.

6) Click OK. Depending on the size of the Media folder, the transfer of files from the former drive and folder to the newly selected one may take hours.

Note that all of the original files will remain in the original location. Once you are satisfied that the files have been completely copied, you may delete the original iTunes folder. But I recommend keeping it for as long as possible, as a backup. And in case something did not copy.