FYI, I did find the Terminal command which would try replacing the current partition table with the backup partition table. Depending on what caused the change in the partitions, this could restore the partition table to the previous layout, or the backup partition table is identical to the current layout.
I would not use this command until you have finished recovering your data just to be safe since I don't know if it will cause any problems with the data recovery software if it is already midway through a scan/restore.
If you want to try replacing the current partition table with the backup partition table, then you first need to get the device identifier for the external drive. You can get this device identifier from Disk Utility. It will be in the form of "diskX" where "X" is a number such as "disk7" as in the command line pictures you last posted. Keep in mind you need to retrieve this device identifier just before using any of the command since the device identifier can change every time you connect the drive.
Once you have the correct device identifier for the "broken" drive, you can use the following Terminal command to replace the current partition table with the backup copy of the partition table (replace "diskX" with the correct device identifier for the "broken" drive):
sudo gpt recover diskX
This command will prompt you for your admin password. Nothing will appear on the screen as you type the password. You will need to press the "Return" key to submit the password.
If the command doesn't produce any errors, then eject the drive and reconnect it so it will pick up the other partition table.
