Andreas is incorrect here. A double spend _did_ happen.
Bitcoin's double-spend protection is probabilistic: after one confirmation, if the sender is attempting to double spend, the probability of success is extremely low. But it's still non-zero.
As the Bitcoin whitepaper helpfully explains, the probability of a double spend attempt succeeding drops exponentially with the number of confirmations. One confirmation is extremely low, two is basically (extremely low)².
Full details: https://bitcoin.org/bitcoin.pdf
Unfortunately, Andreas falsely claiming a double spend did _not_ happen is dangerous: people have to realize that a single confirmation is not an absolute guarantee.
This case was ~$20, and looks like someone was just moving money between different wallets.
But if you're accepting a payment large enough that you can't risk even a very low chance of a double spend, you _do_ need to wait for multiple confirmations. Claiming otherwise could lead to people losing money.
As an example, my OpenTimestamps server software waits 5 confirmations (by default). I'm not worried about losing money - the calendars are double-spending transactions to themselves. But the software frankly can't handle a double spend - you'd need to manually fix things in the calendar database.
So I just wait a few confirmations to make the probability of a double spend extremely low. So low that if one actually does happen, it's much more likely that Bitcoin itself broke in some way.
Re user's being confused, here's an example.
Even with control of no hashing power at all, you can very occasionally do double spends of confirmed transactions. You just have to be lucky enough for a stale block to get mined at the right time.
Stale blocks are pretty rare these days. I don't have stats handy. But looks like the % is something like 1 in 1000 blocks or less. So that'd be a sub 0.1% chance of success per attempt.
Thank you very much @pete for this awesome and complete explanation