iMovie - Problem with external drive showing up twice

what options does right clicking give you ?

Exit iMovie.
From the Finder's"Go" Menu:  Type /Volumes/ within Go to the folder:

From there you will see your Hard Drive, any currently mounted servers/disks and aliases to old disks. Select any/all aliases that don't apply and throw them away.

Open iMovie. Should show only valid drives.

Here's a screen shot of right click
Screen-Shot-2013-11-11-at-9.56.0.png

Re: ultralites comment (thanks!)

I had tried your advice.  "Volumes can't be found" (see screen shot).  Perhaps I should highlight the drive I'm having problems with before I type "volumes"?
Screen-Shot-2013-11-11-at-10.26..png

needs the slash before and after: /Volumes/

Ah, thanks :-)

So now that I see both "E" drives...here's my thought? Delete a whole drive? What if it wipes all the information out on that drive?  All my family pictures?  Is there a way I can test this without committing to a delete?

Any REAL drive will have the eject button next to it.

You should only delete the drives that are using an aliased icon.
Any/all alias icons will have a curled-up bottom left edge or an arrow.

Both drives have the eject button (see screen shot).  

When I expand the drives, they both have the same contents.
Screen-Shot-2013-11-11-at-11.15..png

After a disconnect and reconnect to "Ranier", I see only one E drive - both in Finder and iMovie.  Thanks for your help - it's very much appreciated!