This is a tough one, but I'm going to weigh in on this needing an Attack type. The non-attack interactions are all pretty neutral. Tribute cycles somebody, and that might be a good thing (skipping Curses) or a bad thing (skipping Minions), but it's not clearly better or worse in a generic situation, and -- this is the key -- without some very particular deck inspection combinations, you can't really use Tribute as a de facto attack.
Possession is a blurrier case, but I believe the same can be said of it. The norm is that it cycles the opponent a bit and swaps a random hand out for a different random hand -- no harm done on average, and you can't really use it to cause harm outside of certain specific combos. Admittedly, Possession, unlike Tribute, has some KILLER combinations, such as with Ambassador, Masquerade, and Island. That's why I say it's a blurrier case. But on its own, it doesn't really harm the opponent so much as help the player.
Masquerade is blurrier still, because it's not really a neutral effect. However, it's just as likely that he'll be able to get rid of a worse card than he gets back, so -- again in the absence of specific combos -- it's a little bit hard to use it to cause any real harm except by accident. But I think the real reason it's not an attack is that the actual mechanics become a bit more complicated if it were possible to Moat it. Not unworkable, but certainly more complex. So let's give Masquerade the benefit of the doubt.
But swapping your entire deck out for a pack of Silvers isn't a neutral action by any remotely conceivable stretch of the imagination. Absolutely it might help you sometimes! But Thief and Pirate Ship can help (trashing your Coppers), Oracle can help (letting you reorder the top of your deck), Margrave can help (by drawing a key card), Mountebank can help (by juicing up Counting House), Militia/Cutpurse/Torturer can help (by empowering your Library), and so on. The difference is, the effects of these cards are not neutral. In short, they:
(1) Cause a tangible effect on your deck.
(2) Can be employed or withheld strategically by an opponent interested in harming your deck.
Both of these things are true of Exile Isle. You're right that very often it will improve an opponent's deck, but presumably an opponent isn't going to do a swap if it does. And in any case, we're not talking about a little simple cycling or junk-for-junk trade. We're talking, "Hey, that deck you were building up? TOO BAD! Have some Silvers instead!" That's a more disruptive effect than any official card can have on an opponent's deck. On that basis alone, it should be labelled an Attack.
And in those cases where being hit with it would be a good thing, you can always decline to Moat it. There are plenty of Attack cards already where you sometimes want to do this.
