Thr Bank of England seems confused to me

The bank is continuing its usual pitch about the loanable funds model, that is, for every loan there is an equal deposit.  They say:

Lending starts with banks collecting deposits of real savings from one agent, and ends with the lending of those savings to another agent.

Actually no, deposits and loans  are collected together, never the one before the other. The model I use presupposes competition between the savings sand loan queues for transaction space and they are independent flows.

In the real world, the key function of banks is the provision of financing, or the creation of new monetary purchasing power through loans, for a single agent that is both borrower and depositor.4  

Not true, in fact this is the misunderstanding that causes major recessions.  The currency bank shifts liquidity  between loans and deposits with interest charges and in a closed economy that means the total outstanding lent money if a bounded variable.  The central bank must both create and destroy loans and deposits.

Specifically, whenever a bank makes a new loan to a non-bank customer X, it creates a new loan entry in the name of customer X on the asset side of its balance sheet, and it simultaneously creates a new and equal-sized deposit entry, also in the name of customer X, on the liability side of its balance sheet. 

The lender is always pre-qualified and take the loan as cash in advance, the currency banker is competing with depositors to supply the money, in essence. The key here is that neither borrowers, depositors and banker have complete, on demand, access to the trade book; access is implicitly priced.
Deposits do not requal loans except in the illusion of the Bank of England. In the real world the shadow bankers are constantly correcting the Bank of England, maintaining the natural yield curve.

The debate is from the monotonarians who claim a government monopoly on money. That is a conceited view and the cause of many monetary failures.