So "pudding" is a generic word to describe a sweet course after a savoury one - an alternative word would be "dessert" or just "sweet". In my family, we would say "would you like some pudding" referring to another course, rather than any specific dish.

And then pudding can also refer to different types of pudding: we also have creamy puddings, like rice pudding (one of my favourites), and things like suet pudding and sponge pudding, which could be baked in the oven or steamed over water on top of the stove. Suet pudding (there are savoury and sweet versions) was often rolled in a cloth and tied for steaming, but sponge puddings (and Christmas pudding) are cooked in a pudding basin, which has a lip at the top, so you can cover the pudding with a greaseproof paper lid, tied around the basin with string. You make a little string handle, so you can lower it into the steaming pot.

These are sometimes considered old-fashioned puddings now, and sometimes referred to as "nursery food", but I've found that visitors to England love them!