The question is: how do you define different bank accounts? Let me use a couple of examples here to help clear up the problem. Let's just hypothetically say there are only 4 different types of bank accounts and for this example they are:
normal everyday account: low interest, with fees, no minimum balance and used for transactions such as paying bills and receiving your pay
savings account: higher interest, used mainly for putting aside money for savings, low or no fees, possible minimum balance (maybe 5000 euro, maybe none), can also be used in the same way as a normal account
checking or cheque account: same as a normal account but is also possible to write checks/cheques to pay for things and may also be possible to overdraw the account at very high costs
credit card account: the type we all know as Visa, Mastercard, American Express etc.
Supposing these are the 4 main types of bank accounts, a Girokonto is really no different to a normal bank account except that it has special features like high interest and no fees and therefore is in fact a savings account.
I have no idea what a "current" bank account is except for saying that it is the one you are using right now, at the moment, your "current" account, it would be like saying in German ihr "Aktuellekonto". I'm a native English speaker from Australia so that could be the reason I've never heard of it before but we use a lot of the same language as the Brits so I don't understand it. Savings account is the most commonly used term for a "normal" or "everyday" account.