Narrow banking would restrict banks to holding liquid and safe government bonds as opposed to other equities (like loans) against depositor's money as opposed to other assets (such as gold as in the case of the Texas Bullion Depository or cryptocurrency as in the case of proposed banks like Custodia [1]).
Making private loans or holding other depositors would be made by the other financial intermediaries along with only holding depositor money is what separates such banks from full-reserve banks.
Some early thought leaders in narrow/safe banking include: In 2019, the Federal Reserve denied approval for such banks in the U.S., claiming that they would: interrupt implementation of monetary policy, threaten the repo market, and that the bank is 'too safe' and would thereby threaten general financial stability.
[5] The Mit Ghamr Savings Bank in Egypt ran from 1963-67.
Islamic banking and finance requires tying financial transactions to real assets.