For clarity of intent, the wording on I Drink Your Milkshake! is as follows:

You may search an opponent's deck for a card. If you do, remove it from the game face-down. For the rest of the game, if that opponent plays a card, you may reveal the removed card if it has the same name. If you do, you may play it for free even if you cannot meet its threshold. If you play it, terminate the opponent's card. (Playing a resource does not trigger this effect.)