The testers a typical jeweler or diamond merchant might use only identify between diamond, cubic zirconia or moissanite. They do not identify origin of a diamond.

The testers that can identify a diamond's origin are quite expensive and usually only found at major gemological labs.

All synthetic white diamonds are "Type IIa" (no or trace amounts of nitrogen) while only 1-2% of mined diamonds are IIa. 97% of mined diamonds are "Type Ia" (clusters of nitrogen). It is possible that is what they were referring to about purity, but the amount of nitrogen is not a definitive indicator of origin.

Perhaps the same documentary, but a while back I saw something where the latest techniques were easy to spot for being too good. But then they showed a new technique being developed that was able to inject natural seeming imperfections. The show claimed that this was the impetus for DeBeers laser etching their mark on diamonds as they had no other way of detecting the difference.

This was a while ago though so I'd imagine the cat & mouse game has evolved for both sides.

That's how all synthetics are identified... natural diamonds all contain impurities from the material they were formed near/in.

It's sort of funny in a way... impure diamonds are worth more than pure diamonds... but then again, none of them would be worth much at all if DeBeers didn't control the release into the market (diamonds are not rare at all, and therefore only valuable due to artificial scarcity).