I see what you mean by Trader, but I still don't see what you mean by Citadel playing a card. You said: "consider a Smithy set aside with Summon. Citadel is able to put it into play." Surely the thing that's able to put the Smithy into play is Summon? Why is Citadel playing a card that Summon is trying to play? And how does this lead to a Captain'd Smithy going into your play area?
Clearly Citadel triggers when you're about to play a card. It can't trigger after you already played it (like old Citadel did), because then it can't play it twice, which is the whole point of the errata.
So it triggers before you've actally played the card. When you "would play" it. And it's able to put the card into play even if it was not in your hand. That's why I wrote, "it expects it to be where it was when you were going to play it." You were going to use Summon to play a set-aside card, Citadel kicks it and plays it (twice) instead. When Citadel triggered, you were going to play a card from Summon's set-aside-land. That's the card Citadel plays instead, from that location.
Likewise, Citadel plays the card that Captain was going to play. But Citadel just says "play it twice", not "play it twice, leaving it there".