Grendel,
I had checked that page a million times - however, with your suggestion I went back one more time, and ta-da, I figured it out.
Previously, I removed .mov from musicbox_filetypes_quicktime in settings. However, LDM still considered .mov files (possible bug?) a Jukebox media file. This meant that it used the can_play_musicbox and can_save_musicbox permissions, both of which I had not set it up for anything other than administrator. I fixed the problem by enabling Unregistered and Registered users in can_save_musicbox.
So, Grendel, your advice saved the day. Thanks!
Edit - just saw your post Andrew, it came as I was writing mine. As you can see, I got it fixed. Possible bug though? Thank you for your reply!