Hopefully the slots are labelled correctly in-game "Officer Stash," etc., but for clarity:
The first slot "The Stash" is officers-only in terms of withdrawal. Valuable things that you may want later access to someday in the future, or things you want given to someone who desperately needs it, can go there. People can message officers to withdraw things from the stash.
The other two bank slots are a bit of a free for all. Anyone can deposit and withdraw. They essentially become a bit of a trash dump for extra items people want to get rid of. Anything you deposit into those slots should be considered communal and up for grabs, but you can typically reliably use it to swap items between alts. Thus you've been using it correctly, except for that first time- sorry about the copper ore if you lost it for good
Every so often, an officer will consolidate and sell off the items in these large slots to convert to silver and then deposit bank into the bank for members to use (edit: or, to purchase needed Influence, which powers buffs- per below
) if they are in need. Jira (an officer, of course) was probably doing exactly this whenever Zinotaru was voicing his concern.