While typo's write-up is a technically correct definition of what people mean when they use "double opt-in" I was surprised to see that one important fact about this term was not mentioned: "Double opt-in" is generally considered to be obfuscatory spammer-speak for what the rest of the world knows simply as opt-in.
On the Internet, nobody knows you're a dog. Neither do they know that the e-mail address you're giving them is really and truly yours unless it is verified somehow. A mailing list that does not take the security precautions typo's w/u describes (aka an open list) isn't any flavor of "opt-in" because the term "opt-in" denotes choice. Open lists effectively turn a blind eye to the (very real) possibility that the people who own the addresses being subscribed are not actually getting a choice in the matter. Because of this vast potential for abuse it is extremely naive to think that an unverified subscription request somehow counts as a "single" opt-in.
Spammers and spam supporters (such as the Direct Marketing Association) use the term "double opt-in" because it sounds impressive and implies that plain vanilla "opt-in" refers to an open list. Because anti-spammers are always endorsing opt-in this legitimizes open lists in the eyes of people who don't know what opt-in really means. Spammers love to run open lists as it gives them an excuse to spam. When the heat comes down they can simply claim that Hacker X forge-subscribed a Millions CD and there's nothing they can do now except offer opt-out because their formerly pure as the driven snow list has been polluted.