Duplicate, Groundskeeper, Academy, etc, are in another category. They only trigger when a specific kind of card is gained. They never look back at what a card was.
But since an ability may not become available to trigger (indeed, may not even exist (e.g. with BoM)) until long after the triggering gain (as a result of other abilities triggered by that gain), checking the trigger condition at that point necessarily involves checking the gained card at the time of the gain, which is in the past. Right?
Ok, I see what you mean. I don't actually think what you're describing is possible; I don't think it's possible to gain a card, and as a when-gain from that make a when-gain on another card be available. Remember that Innovation can only be used on the first card you gain. But nevertheless I get your point. The rule is that we keep checking triggers, so the trigger apparently has to look at what card was gained, not the card's current state.
Let's compare two simple scenarios:
1) We gain Band of Misfits and play it via Innovation as a Duplicate.
2) We gain Overlord and play it via Innovation as a Duplicate.
In (1), we may call Duplicate to gain a Duplicate. The trigger is gaining a card costing up to $6, which we did. Then we resolve it, and look at the current state of the card.
In (2), it doesn't trigger, since we didn't gain a card costing up to $6.
So yes, I agree that the triggers look at the past. It means that the triggering event never changes. If we gain an Overlord, that's the only triggering event we have for the whole when-gain window. It's still a bit different from Ritual, Treasure Map and Smugglers, because those cards specifically refer to cards in the past, while this is true for all triggers on all cards.