Generally, yes, they are the same. The term "lead storage battery" is a bit incorrect, but it conveys the idea that the battery in question is a lead-acid storage battery. These storage batteries are the ones we commonly find in vehicles.
With sulfuric acid and water in the electrolyte, and lead and lead dioxide for electrodes, this battery, while having a low energy-to-weight ratio, can deliver some very high surge current, which is what the doctor ordered to start said vehicles.