If they are sitting on excess cash, they can lower interest rates on deposits.

The point is that when banks receive deposits, in the short term, their assets/liabilities doesn’t change.

Sure, but you don't get a useful model of a bank by only looking at the short term like this.

> The point is that when banks receive deposits, in the short term, their assets/liabilities doesn’t change.

They do change: Cash is a (risk-free, modulo safe storage) asset, the deposit is a liability. This has consequences for all kinds of metrics vital to the running of a bank.

Correct, although if they don't do something to make money with the cash they take in as deposits, they won't have sufficient income to cover their expenses, let alone make a profit.

The effecf of the deposit is separate from the effects of decisions the bank makes after the deposit.

The deposit creates a cash asset and an equal liability for the increases account balance.

Other transactions have other effects.