That's the thing here. We're so used at the concept of money that you described that it's very hard to reconcile with what money really is.

You can take any introductory banking system course and it will tell the same story. (I recommend Economics of Money and Banking by Perry G Mehrling)

It's not really a finished exchange.

Money represent a debt.

It's a one sided exchange against the promise of the completion of the other side of the exchange in the future (by someone else in this case).

It's like the amount in your bank account is not really money. It's a debt from the bank: the promise that it will deliver this amount as cash if you ask. (M1)

And money is so liquid that we never think about it that way. Except during hyperinflation periods. And bank account are so solid that we think about them as cash. Except during bank runs.

As a side note, the theory that barter really meaningfully took place is being challenged as there is no real evidence of this.