You do not have the write permission in your operating system to the location on the Seagate drive. Change the permissions on this drive. (It is also possible that the drive is not formatted properly to allow Lightroom to write to it, what operating system and what drive format?)
So far, I've run the LR script for repairing permissions, and verified permissions on the backup drive to read/write.
I am not familiar with this script, however you do not have to verify the permission on the drive. You need to verify the permissions on the exact destination folder where you told Lightroom to copy the photo files to.
What exact version of LR are you using? Go to Help > About Adobe PS LR and post the exact version number.
The Move and add button are grayed out because LR won't Move images from a Memory card, it is best to format the card in the camera to remove the images from the card, and Add because LR knows it is a Removable drive and when removed from the card reader your images would show as Missing.
LR is a Database program. None of your images are IN LR. They are referenced into the Database file, the Catalog file, from where they are stored on your hard drive.
Well, this issue is solved!
The problem definitely revolved around disk permissions.
dj_paige suggested that I reassign the disk permissions of the target external drive (which I had already done, then re-checked a few times). However, on reading the recommendation, I think I misunderstood and checked the permissions on my macbook's hard drive (which were set to read-only). After correcting this, all is well again. It seems odd to me that the permissions on the Macbook hard drive matter when I'm trying to save files to an external Seagate drive (?). Anyway, it does what it's supposed to now.
I think I need to spend more time figuring out how LR works!
As a matter of interest, I think that the problem may stem from the last El Capitan update that I installed. I've been using LR successfully since November without any permission problems.
The LR version I have installed is CC 2015.8 (1099473), OSX El Capitan 10.11.6 for those interested.
If you guys think there's anything else I need to know about this, please advise!
Based on lots of reports here, Mac OS upgrades frequently screw with file system permissions that should not be messed with.