Once a transaction is added to a block, it's allmost impossible to doublespend the inputs, but as long as the transaction is unconfirmed, it's pretty easy to do so.
My best guess is that the person who wrote the wiki meanth that once your transaction is confirmed, it's protected against double-spending.
A successful doublespend is when two transactions with conflicting inputs are both included in a block.
The way i always understood the concept, double spending means only one of many transactions spending the same inputs is included in a block, the other one is cancelled. This is what makes it dangerous. You think you have received funds from somebody, but a completely different transaction spending the same input is included in a block, and your transaction is cancelled, hence, you are no longer payed.