Really banks can name their accounts as they like provided the name isn't misleading. In practice it isn't always clear whether an account is current account or a deposit account. Both types can pay interest - in fact some current accounts have high short term rates to attract new customers.
Current acounts are usually accounts which allow payments from the account by cheque, direct debit cash card etc and instant withdrawal is available etc.
Deposit accounts tend to be for longer term savings and may require notice to be given for a withdrawal; and they generally don't allow regular payments out such as DDs or offer cheque books