The reason the three of us (and more) got the wrong idea was because Apple's ad was misleading (I would bet intentionally so, to avoid a mad rush of downloads as the 1 billion mark approached).
Here is the story by Macrumors that has a screenshot of Apple's ad. The ad says:
"Download an app and you'll automatically get the chance to win a $10,000 iTunes Gift Card, an iPod touch, a Time Capsule, and a MacBook Pro. Just go to the iTunes Store, browse the App Store, and download your best app yet."
The subhead under "We're about to hit a billion" says essentially the same thing as the main copy of the ad, so you are twice given the idea that all you need to do is download an app to win, with no mention that it's specifically the 1 billionth app (or corresponding no-purchase entry) that would win. Tim Cook mentioned in the financial results conference call that they were "just hours away", so I'm sure if the ad were clearer they would have have more people trying to download the past couple days.
I'm not a legal expert to know whether you can imply or leave an important detail out of a contest ad and leave it to the "fine print", but I think it's clearly misleading on Apple's part. They were very clear about it with the iTunes contests they had before, so there is no excuse for them not to do so this time around as well (unless as I'm suggesting, they didn't want the servers to take a hit or go down from all the traffic that would have been generated as the 1 billionth download approached).
OMG a 13 yo could figure it out...