What is the ruling on dealing with non-Islamic banks in what is called a "day-by-day account," and what is the ruling on benefiting from savings account interest after knowing it is forbidden, and what is the ruling on dealing with Islamic banks which are rumored to be similar to non-Islamic banks?

You have done well by canceling your dealings with usurious banks and postal savings accounts, as they involve usury. Regarding the "Day by Day" account: if it invests money in a permissible field through mudarabah (profit-sharing partnership) with its conditions, then there is no objection to subscribing to it. However, if it invests in something unlawful or lends with usurious interest, then it is not permissible to subscribe to it. The current account in non-Islamic banks does not yield interest; rather, it is solely a deposit for safekeeping. Beware of deceptive titles and seek out an Islamic bank or a commercial project to invest your money.