Thanks for clearing that up. I need to get my information from better sources..
Much of the confusion comes from the fact that it has become common to use the words "double spend" to mean something other than an actual double spend.
One of the truly innovative things that was accomplished with the creation of the bitcoin protocol was that it provided a decentralized way to make it impossible to double spend a digital asset. Prior to the creation of bitcoin, it was possible to use a digital signature system to provide a chain of custody proof that control of a digital asset was being transferred from one person to another. However, it required a centralized "clearing house" to make sure that the digital asset wasn't duplicated and control of multiple copies of it wasn't turned over to multiple recipients.
Without the centralized clearing house, a malevolent entity could make an exact duplicate of the digital asset and then "spend" both copies. There would be no way to determine which was the "valid" one, and which was the "invalid" copy. Each recipient would receive an equally "valid" instance of the asset.
The bitcoin protocol uses a blockchain with a proof-of-work to create a public record of which transaction is the "valid" one. Any other transaction that attempts to transfer control of the asset is "invalid", and this can be determined and proven by everyone that has access to the blockchain. Nodes do not trust the information they receive. Every full node on the network validates every transaction they ever receive (and reject invalid transactions regardless of where they come from). Every full node on the network validates every block that is ever receives (and rejects invalid blocks regardless of how much hash power was used to create it). One of the "rules" of the protocol that is used to determine if a transaction is "valid" is that it cannot spend an input that has already been spent in an earlier block in the blockchain. One of the "rules" of the protocol that is used to determine if a block is valid is that it cannot include any invalid transactions.
So, with a large enough share of the global hashpower, you could choose which of multiple transactions is the "valid" ONE. You caould even go back a few blocks and remove a previously "valid" transaction, replacing it with a different transaction that spends the same inputs. However, at any given time, you could only include ONE transaction in the blockchain that spends any given input, and during the time that any particular transaction is in the blockchain, it is the ONLY valid transaction spending that input.
Therefore, it is impossible for 2 (or more) transactions which both attempt to spend the same input to both be simultaneously in a "confirmed" state. It will ALWAYS be publicly clear which transaction is the "valid" transaction at that moment.
What CAN happen with enough hash power is theft and fraud. The entity with control of a large enough percentage of the global hash power could create a transaction to purchase a product or service. Then they could wait for that transaction to receive enough confirmations for the merchant to feel comfortable providing the product or service. Then they could go back and unconfirm the original transaction and in its place confirm a new transaction that sends the bitcoins to an address under the thief's control. As such, they will have taken possession of the product or service without providing the agreed payment. Although you may frequently hear people mistakenly describe this as a "double-spend" (possibly because there are two different transactions being created), it isn't truly a "double-spend" since the first transaction stops being valid when the replacement transaction is confirmed.
Another thing that can happen is that someone can broad cast multiple unconfirmed transactions that each share the same input. Although only one of these transactions will make it into the blockchain and be confirmed, until that happens it is impossible for the network to know which is the "valid" transaction. If a merchant is willing to provide a product or service prior to a transaction receiving its first confirmation, then it can be possible to get the merchant to see the unconfirmed transaction that pays their address while getting a significant percentage of the global hash power to see an unconfirmed transaction spending the same input paying the thief's address. This will occasionally be described as a "double-spend attempt" (since there are two competing transactions that are both "attempting" to be confirmed). Only one of the two transactions will ever be confirmed though, so it isn't an actual "double-spend".
