I think that when the developers started implementing RBF, they used a very misleading word as a verb to this procedure. Double spending is prevented using a peer-to-peer network as the whitepaper says, but they chose that phrase as the proper one for RBF. Maybe a better formulation would be re-broadcasting or broadplacing?

Anyway, in Bitcoin you can't double-spend. You may be able to increase the fee by re-broadcasting your transaction, but that won't mean that you double-spent it.

As for the site, it seems way too suspicious. Try electrum as Charles-Tim mentioned.

I agree that it looks shady, but RBF isn't a scam and it's different than accelerating a transaction. It's more practical to pay for a higher fee than forcing a service to accelerate your transaction.