You are right that the order shouldn’t matter. This question was asked back in the Menagerie previews:
http://forum.dominionstrategy.com/index.php?topic=20142.msg826410#msg826410It was answered that the Throne would be discarded, and there was no mention of the order of the choices.
It reads to me like this wiki entry came from that post (not that this isn’t part of the FAQ from the rulebook, but “other rules and clarifications” which generally come from forum discussions. I think the entry is just a bit unclearly worded.
HOWEVER: There is a strange argument to be made though for why the order might matter, under the latest rules (see the errata thread pinned here). If you choose the Turtle first, then when you play the card the second time, Throne Room fails to put it into play. The rule for cards such as new Band of Misfits says that when you play a card that isn’t put into play, the BoM stays in play for as long as the card it played would have. However, Donald said that “These rules apply to all of the cards that play cards without putting them into play: currently, Band of Misfits, Overlord, Inheritance, Necromancer, and Captain.” So that suggests that it does NOT apply to a Throne Room playing a set-aside duration. But I could see it going either way technically.