Even today, textbooks on Money, Currency, and Banking are more likely than not to begin with an analysis of a state of things in which legal-tender ‘money’ is the only means of paying and lending. The huge system of credits and debits, of claims and debts, by which capitalist society carries on its daily business of production and consumption is then built up step by step by introducing claims to money or credit instruments that act as substitutes for legal tender... Even when there is very little left of [money's] fundamental role in practice, everything that happens in the sphere of currency, credit, and banking is construed from it, just as the case of money itself is construed from barter.
Historically, this method of building up the analysis of money, currency, and banking is readily understandable… Legal constructions, too, … were geared to a sharp distinction between money as the only genuine and ultimate means of payment and the credit instrument that embodied a claim to money. But logically, it is by no means clear that the most useful method is to start from the coin—even if, making a concession to realism, we add inconvertible government paper—in order to proceed to the credit transactions of reality. It may be more useful to start from these in the first place, to look upon capitalist finance as a clearing system that cancels claims and debts and carries forward the differences—so that ‘money’ payments come in only as a special case without any particularly fundamental importance. In other words: practically and analytically, a credit theory of money is possibly preferable to a monetary theory of credit.Perry Mehrling quotes this passage at the start of his essay Modern Money: Credit or Fiat. If you're someone who worries about the vexed question of what is money anyway, you will benefit from the sustained intelligence Perry brings to bear on it.
Readers of this blog may not be familiar with Perry's work, so let me suggest a few things. The Credit or Fiat essay is a review of one of Randy Wray's books, but it makes important positive arguments along with the negative criticism of MMT.  A good recent statement of Mehrling's own views on the monetary system is The Inherent Hierarchy of Money. Two superb essays on monetary thought in the postwar neoclassical synthesis are The Money Muddle and MIT and Money.  The former of these coins the term "monetary Walrasianism." This refers to the idea that the way to think of a monetary economy is a barter system where, for whatever reason, the nth good serves as unit of account and must be on one side of all trades. This way of thinking about money is so ingrained that I suspect that many economists would be puzzled by the suggestion that there is any other way of thinking about money. But as Perry shows, this is a specific idea with its own history, to which we can and should imagine alternatives. Finally, The Vision of Hyman Minsky is one of the two best essays I know giving a systematic account of Minsky's, well, vision. (The other is Minsky as Hedgehog by Dymski and Pollin.) Anyone interested in what money is, what "money" means, and what's wrong with economists' answers, could save themselves a lot of trouble and wrong turns by reading those essays. 
But let's talk about the Schumpeter quote. I think it is right. To understand the monetary nature of modern economies, you need to begin with the credit system, that is, the network of money obligations. Where we want to start from is a world of IOUs. Suppose the only means of payment is a promise to pay. Suppose it's not only possible for me to tell the bartender at the end of the night, I'll pay you later, suppose there's nothing else I can tell him -- there's no cash register at the bar, just a box where my tab goes. Money still exists in this system, but it is only a money of account -- concretely we can imagine either an arbitrary unit of value, or some notional commodity that does not circulate, or even exist. (Historical example: non-circulating gold in medieval Europe.) If you give something to me, or do something for me, the only thing I can pay you with now is a promise to pay you later.
This might seem paradoxical -- jam tomorrow but never jam today -- but it's not. Debts in this system are eventually settled. As Schumpeter says, they're settled by netting my IOUs to you from your IOUs to me. An important question then becomes, how big is the universe across which we can cancel out debts? If A owes B, B owes C, and C owes A, it's not hard to settle everyone up. But suppose A owes B who owes C who owes …. who owes M who owes … who owes Z, who owes A. It's not so easy now for the dbets to be transferred back along the chain for settlement. In any case, though, my willingness to accept your IOUs depends on my belief that I will want to make some payment to you in the future, or that I'll want to make some payment to someone who will want to make a payment to someone …. who will eventually want to make a payment to you. The longer the chain, the more important it is for their to be some setting where all the various debts are toted up and canceled out.
The great fairs of medieval Europe were exactly this. During their normal dealings, merchants paid each other with bills of exchange, essentially IOUs that could be transferred to third parties. Merchants would pay suppliers by transferring (with their own endorsement) bills from their customers. Then periodically, merchant houses would send representatives to Champagne or wherever, where the various bills could be presented for payment. Almost all the obligations would end up being offsetting. From Braudel, Capitalism and Civilization Vol. 2:
... the real business of the fairs, economically speaking, was the activity of the great merchant houses. … No fair failed to end with a 'payment session' as at Linz, the great fair in Austria; at Leipzig, from its early days of prosperity, the last week was for settling up, the Zahlwoche. Even at Lanciano, a little town in the Papal States which was regularly submerged by its fair (though the latter was only of modest dimensions), handfuls of bills of exchange converged on the fair. The same was true of Pezenas or Montagnac, whose fairs relayed those of Beaucaire and were of similar quality: a whole series of bills of exchange on Paris or Lyons travelled to them.
The fairs were effectively a settling of accounts, in which debts met and cancelled each other out, melting like snow in the sun: such were the miracles of scontro, compensation. A hundred thousand or so "ecus d'or en or" - that is real coins - might at the clearing-house of Lyons settle business worth millions; all the more so as a good part of the remaining debts would be settled either by a promise of payment on another exchange (a bill of exchange) or by carrying over payment until the next fair: this was the deposito which was usually paid for at 10% a year (2.5% for three months).This was not a pure credit-money system, since coin could be used to settle obligations for which there was no offsetting bill. But note that a "good part" of the net obligations remaining at the end of the fair were simply carried over to the next fair.
I think it would be helpful if we replaced truck-and-barter with something like these medieval fairs, when we imagine the original economic situation.  Starting from a credit view of money modifies our intuitions in several, as I see it, helpful ways.
1. Your budget constraint is always a matter of how much people will lend you, or how safe you feel borrowing. Conversely, the consequences of failing to pay your debts is a fundamental parameter. We can't push bankruptcy onto the back burner as a tricky but secondary question to be dealt with later.
2. The extension of credit goes with an extension of the realm of the market. The more things you might be willing to do to settle your debts, the more willing I am to lend to you. And conversely, the further what you owe runs beyond your normal income, the more the question of what you won't do for money comes up for negotiation.
3. Liquidity, money, demand, depend ultimately on people's willingness to trust each other, to accept promises, to have confidence in things working out according to plan. Liquidity exists on the liability side of balance sheets as much as on the asset side.
4. When we speak of more or less liquidity, we don't mean a greater or lesser quantity of some commodity designated "money," but a greater or lesser degree of willingness to extend credit. So at bottom, conventional monetary policy, quantitative easing, lender of last resort operations, bank regulations -- they're all the same thing.
When Minsky says that the fundamental function of banks is "acceptance," this is what he means. The fundamental question faced by the financial system is, whose promises are good?
 I don't want to get into Perry criticisms of MMT here. Anyone interested should read the article, it's not long.
 MIT and Money also makes it clear that I was wrong to pick Samuelson's famous consumption-loan essay as an illustration of the neoclassical position on interest rates. The point of that essay, he explains, was not to offer a theory of interest rate determination, but rather to challenge economic conservatives by demonstrating that even in a simple, rigorous model of rational optimization, a public pension system could could be an unambiguous welfare improvement over private retirement saving. My argument wasn't wrong, but I should have picked a better example of what I was arguing against.
 Perry has also written three books. The only one I've read is The New Lombard Street. I can't recommend it as a starting point for someone new to his work: It's too focused on the specific circumstances of the financial crisis of 2008, and assumes too much familiarity with his larger perspective.
EDIT: I removed some overly belligerent language from the first footnote.