Issue moving iTunes external library over to new iMac

Try holding down the Option key while you start up iTunes. That'll bring up a dialog that'll let you select your library on the external drive, or create a new library.
I'd make a copy of the ext drive lib file before doing anything though. That way when the new version of iTunes you probably have screws up while trying to update the library to match current specs, you'll have something to give the process a second shot.

 

I had a similar battle and went through several rounds of thinking I might have solved it only to find that iTunes couldn't see some media later on. Eventually, I arrived at a painful solution of using the old Mac to consolidate the external library into the music folder (where Apple seems to want it to live). Move that (newly consolidated music folder) to the new Mac. Choose that library on the new Mac and make sure it's all working fine (spot check media files). Then, move that library (on the new Mac) back to the external where you want it stored for the long run. Make sure both versions of iTunes are up to date before doing any of this.

I followed online suggestion after online suggestion the other way and could never get it to "just work" in a complete way until I did the above. Since doing the above, it has worked just fine.

If you don't have enough room on the original or new Mac to hold your entire library, this would get more complicated. Where you might go with that problem would be clone (super duper, carbon copy cloner) the internal drive of the old or new Mac (with too little space for all of your iTunes media) onto an external drive with enough space, boot with the external drive, do the above (to consolidate your library into the Music folder on the new, external boot drive), then import it to the new Mac, make sure it's all working fine, then move the library back to the external where you want it stored for the long run.

Big hassle? Yes. But it works.

 

I think HobeSoundDarryl's given you good advice. I've never used an unconsolidated library on an external drive, and can see that that very well may be causing you trouble when you switch Macs. The good news is that once you set up and do the consolidation, you shouldn't ever have to do it again.