In the U.K. we call that a bread and butter pudding, The difference from what we call a bread pudding is the bread and butter one uses buttered slices of bread with a sauce poured over. We call the solid, cake like pudding a "bread pudding". Same language but so different really. The American cilantro we call coriander, egg plant is aubergine, etc. etc.
Here's the recipe for the bread pudding my Mother used to make. It is dense and moist and I loved it. Haven't made one for years though.
12 oz. stale white bread, soaked in water for about an hour then squeezed in a clean cloth (butter muslin?) to get all the moisture out. It is then mixed with a 6 oz.mixture of currants and raisins, 4½ oz.sugar, a little grated nutmeg and/or lemon zest, 3 oz melted butter, 2 beaten eggs and approx just under ¾ pint milk, put in a greased pie dish and cooked at 180C.for 1½ hours.