Are there different types of D battery?

Generally speaking yes, depending on how you define things. There are consumer D batteries (they look like all normal batteries) then there are "Tabbed" batteries, these have tabs on the end to be soldered into equipment (normally rechargeable). Batteries are also made up of different chemical types, and have different amp hour ratings; A D-type alkaline cell is normally 12,000 mAh, while zinc-carbon, also described as heavy duty or super heavy duty types, have about half that.