Part of the reason I linked to the other thread was that it has a discussion about which event "happens" to whom.
You say that Alex chooses which order the triggers happen in, but I'm not convinced that should be so -- it makes sense (enough) to me that the Possession redirect of Province to Alex "happens" to Alex and the Trader replacement "happens" to Betty.
If this is true, then Betty has to reveal Trader first in responding to the fact that she "would" gain a Province
because it's her turn right now -- and because when two reactions/triggers are supposed to happen at the same time, we use turn order as the first way to determine which happens first.
So Betty reveals Trader first. Part of the effect of Trader will be to nullify the gain Province event that was previously hanging in the air. Indeed, Alex's Possession trigger still occurs in response, but by the time it happens, there's no Province to gain...although Alex still gets the Silver Betty would have gained, of course.
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.
I don't agree that if the effect is no longer about to occur Trader can't do anything; the triggering event in the game that sets Trader off is "would gain a card". True, this isn't an event, but it is a stage that can be inferred as part of the gaining process. Once this trigger happens -- once Betty is at a point where she "would gain" a Province imminently -- the Trader can be played.
Obviously, I don't think Possession in this case would trigger before Trader (based on my reasoning above), but even if it did, it wouldn't matter. Here's how I imagine all this going:
1. The triggering event occurs. In this case, Betty declares her intention to buy a Province card, which means she "would gain a card," which is the trigger that would set off either Trader and/or Possession.
2. Now, anything triggered or that may be triggered via Reaction card
is indeed triggered , even if it isn't resolved right away. We need to establish this in order to start determining in what order to resolve the new triggered events.
3. We figure out which order the new triggered events occur.
4. The Possession, if it occurred first, would redirect Province to Alex, and then the Trader could still be revealed. However, in this case, I think that means Betty's Trader would resolve first. This would nullify the event that triggered all this to begin with,
but because we've already gone through step 2, we still resolve Possession's triggered event (the redirect from Betty to Alex).