What is "current rating" in a battery implied by?
It depends if the battery has a protection mechanism or not. For instance, Lithium-based batteries can be dangerous if the discharge current is too high, and require protection mechanisms to guarantee safety. Cheap batteries can be found without such mechanisms, but they represent an hazard and can be illegal.

In some cases, due to internal resistance the output voltage will drop depending on the current that is drawn, therefore limiting the maximum current that can be obtained.

You might find this page at Battery University interesting.