It makes a huge difference. Most likely, it's a percentage low enough to indicate that the 10% threshold is carefully chosen. Otherwise he'd tell us.
This is evidence that he has an agenda to promote (that people are too selfish), and that the facts don't support his agenda (most people actually cooperate). So he omits the facts.
The point is we don't know and can only speculate, just like the students in the experiment who don't have the signal either. It kind of sounds like you want to believe the professor has a bias, and are trying to speculate evidence to support that belief. Even in the article he talks about how picking 6 in the first round isn't viewed as selfish, just that this experiment demonstrates tragedy of the commons.
at 10% (depending on class size) each student realizes that their noncooperation goes a good distance of the way towards that 10%
at say 50% threshold, each student thinks their noncooperation matters less, and thus is even less likely to cooperate
at least that's what I'd hypothesize, I suspect there is an interesting paper somewhere that explores this very phenomenon