When a normal bank makes a loan, it does so against the dollars that depositors have put into the bank.
This is not correct. When a bank makes a loan after deciding the borrower is creditworthy they create an "account" for that borrower and the necessary reserves are deposited into that account from thin air. This increases the M2 money supply by the entire amount of the loan, just as paying down the principle of the loan decreases it.
Banks learned long ago that people rarely ask for their money back all at once, so instead of $1 of deposits supporting $1 of loans, they could do something more like $1 of deposits supporting $10 of loans.
This would suggest some limitations to the ability of a bank to loan based upon its deposits. No such limitation exists. Banks never touch depositors' money, or even consider them to make loans. Deposits represent, as you correctly stated, liabilities to a bank and one cannot "loan" liabilities.
if everyone who got a loan from the bank came in and demanded physical cash all at the same time, the bank would go under because it doesn’t have enough cash to meet the demand.
Cash only represents about 8-10% of the money supply, rising and falling seasonally. Cash is "purchased" from printing and engraving under Treasury as it is needed "as a service" to bank customers. It is treated as a commodity by banking, not currency.
If you pay your taxes in cash at an IRS office it will likely be shredded after your payment is properly recorded. This is because tax receipts are destroyed in the context of being "money" and it is just easier to shred the commodity than do the paperwork involved in recycling it back into federal spending.
So at the end of the day, the underlying amount of actual cash that the bank holds is a limiting factor to how much money it can create (both from a regulatory perspective and a not managing your bank like crap perspective).
If a bank doesn't hold enough reserves to cover 100% of its loans it must borrow them from others that have excess reserves or from the Fed. The interest charged to them is what most people consider as the "prime" rate that the Fed sets to establish a "cost" of money.
Since they do carry an interest charge they are also considered "liabilities" and reduce profits. This is why banks have an incentive to pay them down as loan principal is reduced and to entice longer-term deposits not counted as M2.
So while both normal banks and the Fed can expand their balance sheets (a.k.a. make loans), only the Fed can expand its balance sheet in an unlimited fashion because it never has to worry about actually paying out on its liabilities.
The one thing to keep in mind in making such a broad statement is that only money created by Congress can "net retire" a loan without incurring more private-sector debt. The end goal of a bank is not to gain more of its own credit money, which it can create at will, but to obtain US dollars, public money, from interest. If you pay off a loan you effectively reduce the net money supply by the amount of the principal by balancing the liability of the bank that made the loan with US dollars.
If Congress doesn't "deficit" spend (remember that tax receipts are destroyed money and not recycled) sufficiently to retire bank credit in a timely manner defaults increase and place the entire system at risk. The Fed only acts as the clearing bank for Congress and has no authority to create the public money that is represented in Treasury bonds as the national debt.
Only public (high power) money can net retire loans or be net saved. This is often masked by constantly expanding bank balance sheets and GDP, but doing so creates a fragile economy that falters with each downturn of the business cycle and encourages a bubble/bust economy.
