Following up on two posts ago, my own view is that the most important line to be drawn in discussing monetary aggregates is between the monetary base and everything else. The monetary base (M0) is the medium accepted for final payment within the domestic monetary system. In the United States and most other countries today, it comprises notes issued by the central bank, coins issued by the central bank, and demand deposits of commercial banks at the central bank. When held by commercial banks rather than the public, the notes and coins just mentioned count as part of their reserves, along with their demand deposits at the central bank.
In earlier posts I have stressed the importance of keeping in mind the difference between money in the sense of the monetary base on the one hand and broader measures of the money supply, which are various forms of credit, on the other hand. The public colloquially calls all of them money, and bank deposits, for instance, do serve as generalized media of exchange. The medium of exchange function is not all there is to money, though. Money as opposed to credit constitutes final payment for a good. When a Wells Fargo Bank customer writes a check to a Citibank customer, the transfer of deposit credit constitutes payment as far as they are concerned, but it is not final payment as far as their banks are concerned. Wells Fargo has to transfer reserves (monetary base) to Citibank to the extent that it lacks offsetting claims on Citibank.
There can be complications to the story. Under a gold standard operated by a central bank, for instance, the monetary base is final payment within the domestic monetary system but gold is final payment internationally. That is a wrinkle best left for another time.
Besides the distinction between final payment and less-than-final payment, another reason for distinguishing between the monetary base and the types of credit that constitute broader measures of the money supply is that historically, it has been the case often (much more often than not during the last century) that the monetary base is supplied monopolistically while the types of credit that constitute broader measures of the money supply are supplied basically competitively, though often subject to many distorting regulations such as minimum reserve requirements and government-imposed credit allocations. The credit aggregates are somewhat substitutable for one another in a way that they are not substitutable for the monetary base. One credit technology may entirely replace another, as checks and wire transfers have done to bills of exchange. A credit technology may reduce the demand for the monetary base, as interbank clearing has done, but it is does not eliminate demand for the monetary base unless trust is so perfect that banks are willing to hold one another's IOUs perpetually instead of demand final settlement of debts outstanding.
My reading of the monetarists is that for them, the line between M0 and M1 or M2 is in practice less important than the line between M2 and M3—not neglected, just less important. They consider the statistical correlations between monetary aggregates and economic activity to be highly important, while for me they are less important than the underlying market process—or, in the case of a monopolistically supplied monetary base, the nonmarket process.
I also consider that the Divisia aggregates give a better sense of the substitutability among credit aggregates than making a strong distinction between some aggregates and others, as the monetarists have done, as Murray Rothbard did in America's Great Depression, and as subsequent Austrian efforts inspired by his effort have done.
MISCELLANEOUS: Because of the interest my Bitcoin post aroused, I will collect my thoughts on the subject and write something substantial about it. I have also promised Mike Sproul a post or posts on the "backing" theory of money with right of rebuttal for him and will eventually get around to it. Finally, on the subject I discussed here, J.P. Koning's Moneyness blog has had a number of relevant posts and is worth reading for whatever else he writes as well.
ADDENDUM: In some countries, including the United States, coins today are issued by the treasury as notes were in the past. These details do not change the overall story. Also, the theory of free banking contends that for the monetary system to most nearly achieve equilibrium, the monetary base as well as the provision of credit based upon it should be competitive. I thought that on this particular blog, the point would be obvious, but apparently not so.