To be fair, we're talking about different issues here, because under notice and takedown, it wouldn't matter whether we had the definition you're talking about or not. There's no proof of infringement; you just ask for it to be taken down, and it has to be taken down regardless of whether there's infringement or not.
I would argue that under notice and notice, there is a graduated system. You send a notice, and the person who put on the clip in question, or whatever it is that's in question, gets that notice. They now are aware that someone thinks they may be infringing and they have a decision to make. If they make a decision to keep it up, then legal action can be taken.
We do live in a country where due process is important. I think this is an important question for most families, considering how they share their lives with their friends and the world in general in this new digital world we live in. I think they would have real concern with the fact that if someone says they're infringing, automatically they're assumed to be infringing, and something is ripped off the Internet.