Yes, I don't think it's fair to hold the Bank of England to Graeber's interpretation.

From a few readings, my take away is that the lending of banks is limited not by availability of notes printed by the Reserve (and thus a proportion of savings deposits), but instead by the cost of a bank's deposits, and the return on its assets. From page 20:

Banks receive interest payments on their assets, such as loans, but they also generally have to pay interest on their liabilities, such as savings accounts. A bank’s business model relies on receiving a higher interest rate on the loans (or other assets) than the rate it pays out on its deposits (or other liabilities). Interest rates on both banks’ assets and liabilities depend on the policy rate set by the Bank of England, which acts as the ultimate constraint on money creation.

Thus, the bank can lend when they feel the loan will perform well enough to cover their liabilities, and this lending creates more money as it increases their balance sheet (to be reduced as the lender pays it back).

From page 20:

The supply of both reserves and currency (which together make up base money) is determined by banks’ demand for reserves both for the settlement of payments and to meet demand for currency from their customers — demand that the central bank typically accommodates.

This demand for base money is therefore more likely to be a consequence rather than a cause of banks making loans and creating broad money. This is because banks’ decisions to extend credit are based on the availability of profitable lending opportunities at any given point in time. The profitability of making a loan will depend on a number of factors, as discussed earlier. One of these is the cost of funds that banks face, which is closely related to the interest rate paid on reserves, the policy rate.

That demand that the central bank typically accommodates part is where I think a potato/potato argument can be made. What I think the Bank of London is saying here, is that the Reserve isn't making the decision of whether or not there needs to be more money in circulation, but it is the banks that are making that determination. In fact, the Reserve adjusts the costs of lending, and that alters the math when a bank is deciding whether or not to create a new loan, and this bring more money into the system.

It's very interesting stuff.