So if it is a permanent, the controller decides?
I want to do another thought experiment with a made-up card.
Suppose I want to discard DSC to Thirst for Knowledge on both my opponent's end step and in my first main phase. Now suppose he has a card in play with the following text both times:
"If a card would be put into a graveyard, remove it from the game instead."
Is this the same as the first question?
Yes. You own the affected object, therefore you decide what order to apply the effects. It's all here:
419.9a If two or more replacement or prevention effects are attempting to modify the way an event affects an object or player, the affected object’s controller (or its owner if it has no controller) or the affected player chooses one to apply. Then the other effect applies if it is still appropriate. If one or more of the applicable replacement effects is a self-replacement effect (see rule 419.6d), that effect is applied before any other replacement effects. If both players have to make these choices at the same time, follow the “Active Player, Nonactive Player rule” (see rule 103.4).