Pretty much every card is luck-based. Terminal draw relies on drawing it along with Villages, for example.
This is an interesting point and got me thinking about why some kinds of luck are annoying and others don't bother me at all.
Bad: Luck where one outcome is (nearly) strictly better than another: I think it's obvious why this is annoying. Think of the sinking feeling when Swindler hits an Estate. Examples: Swindler, Harvest.
Good: Luck that self-compensates: This means that if you get a good outcome now, it encourages bad outcomes in the future, and vice versa. The primary example is drawing, where provided cards don't miss the shuffle, you'll see your good cards eventually. (Note: Dominion is particularly good at this because you draw through your deck several times during the game. A game like Hearthstone, where you typically don't draw your whole deck, does not have the self-compensating property to the same degree.) Cards that break this property of drawing, like Minion, can be frustrating. Examples: drawing, Mystic.
Good: Luck you can influence: The luck needed to match Village and Smithy together is greatly influenced by how you build your deck. Examples: engines, defense against trashing attacks.
Good: Luck that is less relevant in high-level play: For example, unassisted Treasure Map has some chance to do very well. But since the probability is unfavourable, in high-level play, players won't try that strategy, making it irrelevant. A nice effect of this type of luck is that it increases the game's luck in casual play, where you want more luck than in competitive play anyway. Examples: Treasure Map, Black Market, Possession.
Good: Luck you need skill to put to your advantage: Consider a favourable opening split. This only helps you if you know what opening makes this split favourable to you. This factor can vary among skill levels: a low-level player might not know that using your 5/2 opening to get a City is a bad idea, but high-level players will. Examples: opening split, order of Ruins and Knights piles.