I think the problem with what you're saying is the ambiguous use of the word 'cash'. Deposits, reserves, and physical currency are all 'cash' in some sense, but

I think the problem with what you're saying is the ambiguous use of the word 'cash'. Deposits, reserves, and physical currency are all 'cash' in some sense, but here it helps to be precise. Banks don't 'lend out cash', they create deposits. Deposits are liabilities for banks. When a bank takes on a deposit, it also takes on a corresponding asset: either a loan in its books, or an increase in its reserve account. In practice the reserve ratio is unimportant for a bank: in Canada and the UK the ratio is 0%. The real constraint on a banks ability to create money is it's leverage ratio, i.e. (assets - liabilities)/assets. In reality the formula is more complicated and uses risk weighting by asset class, but that's the crux of it.