A medium of exchange, not an exchange itself. A medium: "a means or instrumentality for storing or communicating information." Money enables the storage and communication of information about exchanges, as posited by your example.

As for deflation: Isn't there a cost to not reshingling your house? You don't have a reshingled house, something you presumably want. You shouldn't just look at the price of labor, you also have to look at the value of what the labor does: the house, reshingled. Compare that to the cost of reshingling the house. (Opportunity cost, in so many words).

I still fail to see the point of offering folks money to do something pointless, nor how it helps stop deflationary deadlock. If there's a deadlock then they'll just sit on that money as before, no?

I'm not sure I understand the deflationary deadlock -- you posit a situation where everyone sits on their cash and their labor, waiting -- waiting for what? Eventually someone is going to stop waiting and start working, eventually the gap between desired state and existing state will loom larger than the value of sitting on that money. In the house-shingling example, eventually you will value someone shingling your house for $X/hr today more than you will value having a not-reshingled house until the price drops to $Y/hr in Z days. When that point comes, you'll stop sitting on your money and hire someone to reshingle your house. Wouldn't a series of such events unlock the deflationary deadlock as the market reaches the new price point?