Does the NYT have a right (before we get into fair use or w/e) to demand that a video be deleted off youtube rather than made private? I thought copyright was (roughly) about limiting distribution, so even if NYT's copyright claims apply to that video, can they control what the OP does with it if its not being published?
If it were uploaded to YouTube servers, and shared across CDN's around the world, would that not be distribution in the finer sense? Additionally, "private" is different than unlisted, and the video can still be shared publicly, just in a limited sense.
