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.
There is federal spending which creates net financial assets by first issuing new bonds in exchange for existing reserves and then spending those reserves back into the system.
There is swapping reserves for bonds either as they mature or as repurchases.
There is federal taxation which destroys net financial assets by draining reserves.
The creation of that money is no different from any other government spending though. Usually people call swapping bonds for reserves "printing money" (AKA Quantitative Easing) or issuing reserves without the Treasury first issuing bonds on the primary market "printing money". But there is very little difference between issuing bonds "out of thin air" and issuing reserves "out of thin air". Most notably, from the perspective of the private sector, the fact that banks can satisfy their regulatory requirements by owning bonds just as well as they can by having reserves (and the fact that they can typically swap that collateral for reserves as required directly with the Fed) means that there is literally no immediate difference between the two.
So the term "printing money" is fallacious -- either that or you must call all bond issuance and federal spending "printing money" but that would make the term far less dramatic.
You're mixing up the fiscal - Treasury, bond auctions, taxes, deficit, debt service - with monetary - central bank, interest rates, reserve requirements, QE.
The Treasury doesn't have the authority to create money [0], and the Government has to pad its balance sheet with debt.
The central bank on the other hand doesn't have this requirement. (They have the dual mandate to keep PCE at 2% and keep unemployment low [at NAIRU].)
Of course this is "simply" a giant technicality, but people find it useful. (Bond auctions function as an important signaling mechanism. Also credit default swaps issued for sovereign debt, and their secondary market is a kind of useful indicator. Plus the swaps, and the interbank "overnight reverse-repos for reserves" themselves are useful instruments for managing/handling/pricing-in/reifying risk.)
Again, you're right that theoretically anytime someone says "i owe you" they are creating money. And you can run a local bank easily with a spreadsheet and a large initial equity (as criminals oft do).
[0] this might not be true technically, as they might be able to mint coins without Congressional approval and need for appropriations (and hence collecting taxes or issuing debt), but the Fed still has to buy the coins so it's again the Fed that's creating the monetary base for the coin ... ¯\_(ツ)_/¯ - https://en.wikipedia.org/wiki/Trillion-dollar_coin
In reality there is very little difference between creating money (reserves) and creating bonds. The notion that private market discipline stops profligate government spending by limiting the issuance of bonds by treasury is proven false by two things:
Firstly if push comes to shove the Fed would never refuse to honour a Treasury payment and secondly that the bid to cover ratios in bond auctions always indicate a higher demand for bonds than the self imposed/legislated constraints of congress would allow.
So the real constraint on spending is legislation, we could just as easily legislate full employment instead of legislating a balanced budget target.
