La Cie backups

It would appear that you saved an alias on the new drive. If there is a small arrow in the lower left corner of the Lacie drive icon, it's an alias. (Small black arrow pointing to the upper right.)

Assuming it's an alias, when you open it, it will show everything from the original location because the alias just points back to the original location. Again, assuming it's an alias, there is actually nothing else on the Lacie drive.

Regarding backup, you don't want to back up the entire Macintosh HD drive, just your user data. This is normally the contents of your Desktop, Documents, Downloads, Movies, Music & Pictures folders. Plus any other folders or files that you saved outside those folders.

There's probably nothing wrong with the Lacie drive. They are good drives with an excellent reputation.