So in theory, Smugglers should also refer to whatever the card is now. However, what it is now is impossible to know. So I guess it means that Smugglers refers to what the card was when it was gained.
Smugglers refers to the card that was gained; it doesn't care at all about what that card is now. It doesn't care what the specific physical card costs now either (if an effect changed the cost of some copies of a card but not others).
Is there a wording difference reason that Disciple doesn't work the same way in regards to gaining the card that was played?
Cards do not get special phrasings to try to arrange for the best possible results when a card turns out to have transformed into another card. It's an exotic situation and as always the game needs to have friendly wordings.
Smugglers has this very basic question of cost. I buy Province with two Bridges out, does the other player get to Smuggle it, or do they need their own Bridges or what. It's so blatant of a question that the Seaside rulebook answers it. So, aside from any wording issues, I have a ruling there that I would really like to stick to. Smugglers as interpreted looks at what happened on a previous turn; pick a card they gained, gain a copy of it, btw it has to cost $6 or less. That's the intention; thus the ruling on cost in the Seaside rulebook. Thus me matching that for non-cost things like "what card is it now."
By default cards referring to "it" where "it" is some card just mentioned, refer to the card as it is now, if the card somehow changes right then, which is not normal but can happen. So Disciple looks at the current card.