If you are planning to devise a banking system for a world without quick communication, historical record show how banking systems used to work.
First thing to understand is that ordinary people would not use banks, nor generally coins. In a given area (village, town), shopkeepers (and sometimes families) would use tally sticks to keep track of who bought what and when. You would go to the baker, take your weekly stock of bread, and they would notch the tally stick. When the baker goes to the mill, the miller records a notch on the baker's tally for each flour sack. And so on so forth. Once a year, generally on St Martin's day, everyone would reconcile the accounts, and settle the outstanding debt. The one who had coins could use them, the one who didn't could use anything to settle the debt - maybe they would give a pig, or anything of value they would have.
Only wealthy people and merchants would use banks. But again, not like we are used to think about it. Let's say I'm a wealthy individual, and I need to travel. I know that at my destination I will need funds to sustain my standard of living. But, I don't want to travel with money, by fear of being robbed. Then, I can go to a bank branch, deposit some value (coins, or a few bullion of precious metal for example). The bank will issue a note, telling the remote branch that it has to pay the bearer of the note the same amount of money in local currency. When reaching my destination, I can then go to the bank branch, give them the note, and withdraw the written amount of money in local currency.
Deposit accounts (i.e., accounts you described where you would deposit money and withdraw it at any time, anywhere) could only work in a local branch, where employee knew who you were were and would keep track of deposit and withdrawals on a local ledger. If you were to go to another bank branch in another town, they wouldn't know who you are and there was no way you could withdraw any money.
Also, keep in mind that until late in the 20th century, there was no government issued IDs. There was absolutely no way to know if you were who you pretended to, unless by chance someone well known in the area could vouch for you. That is why introduction letters existed. For example, if you planned to go to a specific town, you would look around for someone who knew you well enough, and personally knew someone in this town. They would then write you an introduction letter. Upon arrival, you would go to their remote contact and give them the letter. If convinced by the letter, they would they be able to vouch for you, and confirm to locals that you were who you pretended to, because they trusted the person who wrote the letter.
In summary, a world without quick communication is exactly our world a few centuries ago, and you can draw from historical records to devise your system. If you want firsthand accounts of how it worked, Casanova's memoirs for example are a good place to start.