Why does altering STATIC_URL in Django disrupt ADMIN_MEDIA_PREFIX
Which version of Django do you use? Before Django 1.4, the STATIC_URL and ADMIN_MEDIA_PREFIX are indeed different parts. In Django 1.4, ADMIN_MEDIA_PREFIX is deprecated and STATIC_URL is used solely. 
The way you're using to serve corresponding files is important also. You could check the URLs of media files in the admin page: if they are generated correctly but cannot get loaded, it might be caused by mis-configured web server instead of Django itself.
Trying to gain some high level clarity on how Django serves (admin) static files.
When altering the STATIC_URL setting in Django's settings.py to a new location (I'm serving images from a 3rd party server) it seems to disrupt ADMIN_MEDIA_PREFIX and thus stop the Admin media (css, images, js) from being served.
Why does this happen?  I though STATIC_URL and ADMIN_MEDIA_PREFIX should pertain to different parts of the application and not influence eachother.