This is in response to the part of your question that asks about M1 versus M2, although it seems you've more or less answered parts of your own question. M1 is the simplest monetary aggregate and includes items most widely used as a medium of exchange (approximately 85% of household purchases are made using M1 balances); it is defined as follows:
\begin{align*}
\mbox{Aggregate M1} &= \mbox{Currency held by the public} \\
&+ \mbox{Travelers cheques} \\
&+ \mbox{Demand deposits (checking accounts that pay no interest)} \\
&+ \mbox{Other checkable deposits (checking accounts that pay interest)}
\end{align*}
M2 is a broader definition of money that adds to M1 other assets with check-writing features, such as money market deposit accounts, and other assets that can be turned into cash quickly with very little cost, such as savings deposits.
\begin{align*}
\mbox{Aggregate M2} &= \mbox{M1} \\
&+ \mbox{Term deposits (deposits locked up for a period of time)} \\
&+ \mbox{Savings deposits} \\
&+ \mbox{Retail money funds (mutual funds investing in safe short-term assets)}
\end{align*}
Looking at why central banks usually focus on M2 instead of M1 to monitor monetary policy gives us a quick sense for what drives M1 versus M2:
Interest rate changes. Higher rates entice people to switch balances in checking accounts, which pay little or no interest, into savings accounts, which pay more interest. This activity causes M1 to shrink but does not affect M2. Since people can relatively easily spend money from their savings account balances, it can be misleading to focus on trends in M1.
Financial innovations. The dividing line between checking and savings accounts has been steadily blurred with banks getting around the Fed prohibiting interest payments on checking accounts by, for example, creating savings accounts that earn interest but whose balances were automatically transferred into checking accounts when required. This actually led to the definition of M1 being expanded to include such accounts: "other checkable deposits."
Financial deregulation. Non-bank financial institutions such as mutual savings banks, credit unions, and savings-and-loans associations were at one time not allowed to have checking accounts, so their deposits were not included in M1. Current monetary aggregates include deposits at all financial institutions.
As you can infer from the above discussion, the main driver of growth in M1 versus M2 should be the interest rate offered on money substitutes, as long as the institutional structure of where firms and individuals hold their deposits doesn't undergo a significant change.
However, to therefore conclude that that the M2-M1 differential is primarily determined by commercial banks is probably too simplistic. Clearly, one effect of the Fed's massive bond purchases (QE) is to lower interest rates across the yield curve (and therefore more or less equalize the interest rate differential between checking and savings accounts).
Added Later
In fact, the following article shows how a prior episode of QE led to an increase in the growth rate of M1 versus M2: What's Driving up Money Growth?