Thanks - I understand now.
But, if bitbargain.co.uk is not my wallet, then what is it? I can login to it and send and receive coins.
It is bitbargain.co.uk's wallet. You have no control over it. They hold the private keys and keep track of your balance separately in an accounting database that is independent of the wallet. The only purpose of the public address they give out is to credit the person who deposits money in their wallet.