I'm trying to fully grasp the third section of the whitepaper. Quoting (emphasis mine):
The solution we propose begins with a timestamp server. A timestamp server works by taking a hash of a block of items to be timestamped and widely publishing the hash, such as in a newspaper or Usenet post [2-5]. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it.
I understand the double spending problem described in the previous section and how a timestamp server as described in this section solves it. I don't understand why timestamps are needed.
Let's presume that the timestamp server itself doesn't check for double spending. In this case, in order for the payee to validate the transaction, they have to confirm that the coin hasn't been spent in any previous blocks. Timestamps are irrelevant since the order is defined by the chain of hashes.
I considered the possibility that the author uses the word "timestamp" liberally to refer to a block, but the bold sentence in the quote suggests that it indeed refers to real-world time.
 If it does, then the payee doesn't have to care about the order of transactions.
 The term "block" hasn't been introduced at this point, but it does appear in the subsequent diagram.