Yet website like Localbitcoins.com shows this as an incoming transaction that is ready to be credited.
The transaction is real and was broadcasted, probably before the first transaction in the chain (the one doublespending the unspent output) was confirmed. At that point, it was reasonable of localbitcoins to conclude that, in normal circumstances, the transaction could have been confirmed in ~30 minutes.
However, given the fact that the first transaction in the chain of unconfirmed transactions uses an unspent output that's been double spent (and the double spending transaction was confirmed afterwards), this transaction became invalid and the funds should have been removed from your acocunt overview.
It's probably a bug in localbitcoin's accounting system, they should probably remove these funds cause they should not confirm.
Disclaimer: it's early in my country, i hope i didn't mess up when i was following the chain of transactions... I think i followed it correctly, but i have a habit of messing up when i'm tired... If you want to be 100% sure, you should follow the chain of transactions i posted in my previous post by yourself and see if everything adds up.