is design850.com your website?
if not you should ask them to use their custom skin, out of courtesy. If they allow, you will have to allow cross-domain access within your flash file.
If it is your site then the playback skin will be relative to where your html is. I.e. if your html is www.mysite.com/mydirectory/myfile.html and your swf is in that same directory, then in flash flv playback parameters you put :: Custom...swf. Lets say your html is in that same place and your skin is in mysite.com/mySecondDirectory, then in you flv parameters you need to say :: ../mySecondDirectory/Custom...swf.
Relative to your html, not your swf.
Yes, the design850.com is my site.
I have the swf movie file in the http://www.design850.com/herzog/ directory (and obviously the custom skin swf is in that directory as well).
I have tried it both ways of referencing the skin swf as just CustomBlackExternalPlaySeekMute.swf and http://www.design850.com/herzog/CustomBlackExternalPlaySeekMute.swf but neither work.
In what directory is the html in that you are embedding your main swf?