This is absolutely the correct answer. This joke/riddle gets to the heart of what "money is". It's a form of communication, it's merely a catalyst that enables trade.

Another way to look at it: The debt (or the $100 bill) allowed all these different professionals to offer services to each other, without a direct back-and-forth barter. If they had started with the prostitute renting the room with a $100 bill, there would have been no debt.

The reason why the physical $100 is so important (and why the Fed prints money), is that it allows for trade to occur without this accrual of debt. The existence of the debt creates the perception of risk, which prevents further trade from occurring.