Problem creating Mtn Lion Recovery partition on external drive.

Question

USB external.  Newly erased, three partitions: one 2GB for Recovery HD, other two backups of my old SL machine and new ML machine (Mini). Those work and their respective machines fine.  Drive seems OK.
    Used Recovery Disk Asst v. 1.0 to create Recovery HD on the small partition while booted into 10.8.2 (the brand new Mini).  Factory installed Mtn Lion Recovery HD exists on internal drive and it works— when starting with command-R or command-shift-R, not with Option key.     Install of Recovery HD to external seemed to go fine and doing a "diskutil list" in Terminal showed both the internal and external Recovery HD's, properly named as such.   However, when restarted with Option down the recovery partition showed up named the same as my internal— at that point "Macintosh HD" but later when I changed the internal's name, reinstalled the Recovery HD, and restarted with Option key the recovery partition now was named the same as new internal drive name.   Both times choosing the recovery partition resulted in booting into internal drive.   
Was wondering if anyone had a similar problem— and found solution/reason.   Apple Tech Support particularly not helpful/knowledgeable on this subject. 

Answer 1

Hi Baltwo
   Really appreciate the help.   Many thanks.     Finally seems to be working.     Not sure exactly what worked but in case you're interested I'll give you a, hopefully, short summary:
Using DU (after copy/paste of the command you wrote above into Terminal) I cloned the working internal drives RDH to the partition for it on the ext.   Failed— still seemed to work as almost an alias pointing at the boot volume on the internal.
So, erased the external again; didn't partition, did a reinstall of ML from DU while booted into RDH of internal.  Then started from that new ML partition on external and let Setup Asst migrate everything over from the internal to keep Users/Permissions from being a problem.
    When that finished I tried to restart w option key.   Got a recovery HD that was called exactly that but, again, booted into internal boot drive.   Booted into internals RHD just to make sure it was OK and it was.
   For whatever reason again restarted w option key.   This time I got four choices:  the boot volume of internal and external, an ext volume called Recovery HD (the trickster), and another volume called Recovery 10.8.3.   Something new— and, as Bill Murray said in Ground Hog Day, "anything new is good."   So I tried booting into that new RHD 10.8.3.  (Note, I'm still running 10.8.2 on boot volumes.)
  And that worked— booted into Recovery HD but no way that I could figure to see for sure that it was the Recovery HD on ext vs. on internal.   I think I need to activate the Debug menu on the RHD DU so I can choose the 10.8.3 recovery and see if I have the option of Repairing the disk or it's greyed out.  If greyed, then proves I'm booted into it.   Might be a more elegant way to see that but it's the one I know.    Still, I assume I was in the RHD on the external.
  Then made a second partition for the old mac's backup and that initially resulted in the boot volume of the internal backup not be mountable but after a restart or two testing the two RHD's, the mini decided to see all the volumes and things are, for the moment, copacetic.  
  Hope that wasn't too long and boring.    From my experience and reading on the 'net, my general sense is that there are a bunch of variables that all have to be right for this process to work.  Probably less is more and just installing ML on a disk by itself will be the most reliable method, if most wasteful of data storage. 
Regardless, couldn't have done it without you, baltwo.   I don't use this forum enough to know what the etiquette and protocols are but I marked all of your posts as useful, which they were, and, finally, as solving my issue, which they did. 
Thanks again