Anarchy ≠ no rules
Anarchy = no ruling power / no hierarchy / no law (in the sense of "state-imposed" law).
Exactly; halfawake, the largest flaw in your argument is that anarchism necessitates a lack of rules, but it's actually government which leads to a lack of rules (see: North Korea.) Anarchism means no rulers, which means everyone has rules; because the rulers aren't subject to rules, as they invent the rules, the whole point of anarchy is to stop those people from being above the law.
The other flaw is the private island thing; the island is still owned by government, you only rent the island.
I'm more than a little bit confused by your argument here. North Korea, lack of rules? No, North Korea is a totalitarian society, that means they have WAY, WAY too many rules. I know I'm in the minority on these boards since I'm not a libertarian, but even I would prefer anarchy to that kind of society. I don't live there though, thankfully.
In any case, I looked up anarchy and the definitions I found seem to back up my arguments. But I do acknowledge that I was taking the argument to a bit of a logical extreme, I just think the cases I outlined there are the dangers of living in a place where anarchy is the system.
Here's one of the definitions of anarchy, according to dictionary.reference.com: "A general lawlessness and disorder, esp when thought to result from an absence or failure of government." Here's another: "confusion and disorder" Of course, there's also this one, which backs up your logic: "a state of society without government or law."
I don't know why you're stuck on the idea of governments owning all the land in the world. There are islands that are owned by private individuals that aren't part of any nation state. Such a concept does exist. It's just that they aren't that common, so most of them are probably already owned by some rich individual who you'd have to persuade to sell it to you if you wanted to start said anarchy.
No matter where you are in the world, you are subject to the larger nation's laws. If the larger nation says, "get off this island, we're claiming it", you can never have enough guns to defend yourself against them.
Yes, the definitions take two sides; one implies political disorder and chaos, the other doesn't. This is because people use the term anarchy in both ways; the first is to indicate a lack of politics, the second refers to the ideology. The people who make definitions aren't infinitely wise, so it's up to everyone else to figure out what the word means, and there's some discrepancy as to what would occur in an anarchy, so the definitions must reflect that.
Anyways, what I mean by North Korea is, they don't have rules, because the ruler of North Korea can kill you for fun. That is what I call complete disorder; when your ruler is so powerful, you can die for being accused of a crime you didn't do, that's a point in which there are no rules, since it's all up to the guy in charge whether he'll simply revoke that rule he created to do as he pleases, or not; after all, he's in charge of creating the rules. Though you are correct to say that there is a maximum amount of law here, just as well, because very few individuals have maximum power, the citizens have no idea if their ruler will have a mood swing that day; that's truly a point where there are no rules, which cannot possibly occur even without government, as there would never be an allocation of power so great without it. But I agree, I hope never to join them.
There is no real way to start an anarchy, just as you cannot start a religion named atheism. It is the lack of rulers which indicate anarchy, just as it is the lack of God which indicates Atheism; it wasn't something that was created, it's something that exists only in the absence of something else.
Edit: Check out
this book for an excellent intro and argument for anarchy.
