I've seen this exact thing done. I just can't seem to figure it all out.
Lets say I am working on website.com
What I would like to do is take any random youtube video url let's say youtube.com/watch?v=kffacxfA7G4
Have the user replace "youtube.com" in the url with "website.com"
Have the video play within website.com/watch?v=kffacxfA7G4
I want the video to play in an an embedded player within myexample.com, but pull the video to be played from the url.
Take any embed code from youtube, just make the video id to appear dynamic based on your VIDEOID parameter.
<iframe width="425" height="349" src="http://www.youtube.com/embed/<?php echo strip_tags($_REQUEST['VIDEOID']); ?>" frameborder="0" allowfullscreen></iframe>
With old embed code, there are some author restrictions not to play outside youtube. But now with this iframe embed code, most of the videos are working well.