If a Maester with a Tin Link and a Copper Link attached kneels, who chooses which effect resolves first, the owner or the "first player"? If there were one other attachment in play could the owner this to prevent themselves from being forced to discard a third attachment by giving it the trait "Chain" with the Copper Link so that the Tin Link doesn't discard it?

Alexander West said:

Well, they're responses, and as with any response, the controller of the card decides when to trigger it, or whether to trigger it at all.

Remember, the First Player only gets to determine the order of passive effects - and even then, only when they conflict with each other.

As Rat says, Responses are always at the discretion of the controller, both for order, or for whether to use them at all. Imagine how messed up the game would be if you had to point out all the possible Responses you wanted to use, then let the First Player determine the order. That would make event Responses kind of lame.