1) checking account is a special account for bussineses or people like students, it gives them access and liquidity to funds that are greater than a normal deposit account that will have limitations on the withdrawals usually set a a daily limit.
Current account is an account where you can pay in cheques and have automated payments out of you bank "direct debits" in the UK
Deposit account you can only deposit
Savings account you can only save money in them and often withdrawals are limited.
Banks are very competitive and there is often some overlap on what the various banks will offer for each account, so that an account might offer the same as a different account elsewhere from another bank.
A savings or deposit account from one bank may offer some of the services of a checking account elsewhere but the accounts other available services in that account may be limited, you gain on one hand and loose in the other.