Could any standard fee transaction be double spent by using a higher fee?
The original design assumes a common behavior of processing the incoming transaction on FIFO basis. That means the first transaction received by a miner wins and all conflicting transactions received after it will be ignored (no matter if the first one is already confirmed or not).

While this standard behavior is beneficial for the Network as a whole, some "smart" miners might employ a bit different approach: when building a block, always take the transaction with the highest fee, no matter the order in which it was received.

It might seem this is just a marginal misuse of the protocol and there won't be enough rogue miners doing this, so it doesn't matter. Well, unfortunately, there already is (at least) one mining pool that tries to build its business exploiting this idea. Should it prevail, services accepting zero-confirmation transactions will need to reconsider their policy.