There are two types of money:
Government money or the monetary base (MB) which consists of paper dollars, electronic dollars, US notes and coins.
"checking account money" (aka demand deposits). We have more in checking (in the aggregate) than MB and we accept checks as money so they are money. The most narrow definition of the is M1, but a more accurate/broader definition is M2 or even M3.
The process by which both enters the economy is very different!
Let's start with how MB is created. First it is very useful to study the Fed balance sheet to get a conceptual understanding how this works. For the Fed, when they create money it is a liability. To the individual that holds the money it is an asset. Let's monitor a transaction that introduce 100k into the economy. The Fed sees the rate at which banks charge each other for short term loans change from 4% to 4.5%. Say the target Fed Funds Rate is 4%. The Fed tries to restore 4% by adding money to the banking system. So they go to the "Open Market" and find a bank (only a special one called a "Primary Dealer") that will sell them an asset for 100k (treasury REPOs are usually the asset of choice). The Fed creates a 100k liability (Fed deposit) and uses it to acquire the security (now an asset to the Fed). The Primary dealer now owns 100k in electronic dollars (or a 100k deposit at the Fed that doesn't exist before). The 100k represents a 100k increase in the MB. The Primary dealer of course will spend or relend this to the rest of the economy and that is how MB is created.
So how is bank money created? When you deposit say 22k in cash at a bank, they keep 22k in reserves (cash or electronic dollars), and they give you a 22k claim on those reserves. But money hasn't been created yet. It is when a bank issues more claims on reserves than it has, that bank money is created. So if a banker has 232k in reserve assets and 232k in deposit liabilities, but notices that on average all reserves are never completely redeemed. So say they overbook their reserves by selling some to another bank. Say 32k. The bank now has 32k in say treasury assets, 200k in reserves, 232k in deposit liabilities. Money has been created. This is inherently unstable because the bank is issuing more promises than can be kept and central banks work very hard to subsidize this system so there is never a "bank run". Whether they should is another question (it can cause inflation and instability).
Then some clarifications in regards to your specific points.
Banks do not mostly loan directly to the government, but they do so indirectly. A lot of bank money is used to purchase securities from the secondary market, but not so much the primary market. Certainly bank money is used to purchase many other things like home loans.
Federal reserve notes and coins (M0) are not significant to determining monetary policy because they ommit electronic dollars (or deposits at the Fed). The Fed will gladly switch electronic dollars to paper or vice versa, so the difference between the two is not significant. M0 (physical money) + federal reserve deposits (electronic money) = MB all government money.
Practically speaking the banks do not dictate to the Fed how much MB they receive (which in turn determines how much bank money they can pyramid onto this new MB). A modern explanation of how banks works figures in the influence they cause to the "open market" which means to an extent they can determine how much the Fed gives them. This is very complex and unnecessarily obscures sound underlying concepts of how fractional banking works. I wish many economic students are not told about this first as it really confusing things. I'll try to explain.
The Fed's favorite weapon of choice for introducing MB into a economy is through the "open market". They attempt to influence the Fed Funds rate to be an arbitrary interest rate. If they banks charge each other more for short term loans, this rate goes up and they will induce the Fed to create more MB. In a modern system, banks heavily borrow and lend from each other. Your common bank then in turns borrows and lends from super banks (primary dealers), and super banks (primary dealers) borrow and lend to the Fed. In a "modern system" the banks are not concerned about reserves...because they know indirectly they will be provided (the basic concepts of fractional banking are still valid though).
So say a bank sees a great opportunity to acquire a 100k interest bearing asset. They just buy it (unless their credit rating is suspect). And then borrow the reserves they need to buy it from another bank. If reserves are scare, the rate at which reserves are charged goes up, which induces the Fed to add more reserves to the system. This is complex and again obscures very sound technical concepts of how banking works.
If the banks got together as a cartel and fixed the Fed Funds Rate, then yes, there is not limit to the extent to which the open market could be manipulated to provide reserves to bank (well more so than currently). A logical concern is that the Fed is already buying overpriced assets and getting churned by excess/manipulated buy/sell volume which means your average taxpayer is hurt.