The thing is, I don't think guided's example there:
3b. I choose to activate Mint's when-buy condition before Talisman's. I trash all my treasures from play.
4b. Talisman is no longer in play, so I am not able to trigger its when-buy condition.
...is correct. Guided writes that, since the Talisman is no longer in play after you resolve Mint, you no longer activate its on-buy effect, even though it was in play when you bought Mint. But here Donald writes:
The resolution of the first won't stop us from trying to resolve the second. So isotropic has Royal Seal / Mandarin right.
—in other words, even though Royal Seal is no longer in play after you resolve Mandarin, you still activate its on-gain effect, since it was in play when you gained Mandarin. So I think guided is wrong: if you have Quarry and Talisman in play and buy Mint, Mint tries to trash your Treasures, and Talisman tries to gain you an extra Mint, and both effects go through in either order.
So in this case, even though Betty is no longer about to gain a Province, you still activate Trader's would-gain effect, since she was about to gain one and Trader's effect was triggered, but Alex just decided to activate Possession's simultaneous would-gain effect first.
The difference is that Royal Seal is reacting to something that happened. It's a fact, it occurred.
Trader reacts before the effect it is reacting to actually occurs. So if that effect is no longer about to occur, Trader cannot do anything.
Besides, you really can't have it the other way. Suppose the initial card is gained by the possessing player, and now you have the silver being gained. Reveal Trader again. The possessing player gains the silver... and then there's another silver being gained. Reveal Trader again. And again. And again. So, you get the initial card, plus as many silver as you like, up to the whole stack.
Since that's obviously not acceptable, just give up. You can't gain an extra silver this way.