Pudding in its original form was something akin to sausage. Meats, spices, etc, bound up with grains and steamed in a casing.
Haggis, that Scottish treat, is a remnant of these puddings of yore. They were ways to stretch the life of meat and more expensive products, by both preserving them with a lot of spices, and mixing them with fillers. Pudding, it seems, was the Hamburger Helper of Medieval times.
If you visited the US up until around 1880 or so, and if you asked to be served a pudding, you'd get a traditional English pudding, if you got anything at all. Pudding wasn't a popular dish in the US because there simply wasn't the need to stretch the budget as much any longer. Puddings diverged into sausages and cakes, and in the USA we had a period of extreme nationalism where we moved away from things like drinking tea, and the pudding.
What changed that was Alfred Bird, who in 1837 invented a way to make custard (the base component of most puddings) from cornstarch and sugar, rather than eggs. This made the making of a custard, which was a fussy business in the past, into an easier task. With no eggs, the custard couldn't curdle, even if the milk was slightly sour (which was a problem back then, making a custard often turned into making cheese.)
The custard powder had some other properties as well. It stayed stable as long as you kept it dry. Not so much a selling point in England, but in the USA, there were all these people out in the frontier, moving from areas of civilization to ... well ...
the wild west. Since space was at a premium in your Conostoga, taking chickens to give you eggs wasn't as viable as taking along a bunch of custard powder, which could sub for eggs in a variety of circumstances.

Since you had custard powder as a staple in the US, and since "stuff in a custard" is often defined as a "pudding," the pudding re-entered American cuisine.
But the sharp divide between pudding as a British concept, and the American idea of "all custards are puddings" didn't happen until the beginning of World War II, when rationing meant that dessert was hard to come by. General Foods had a big hit with Jell-O powdered gelatin during that time, and many variants of Jell-O, like the custards of old, contained bits of other foods to "stretch" the dessert. Right after the war, General Foods purchased Alfred Bird's instant custard company, and they incorporated Bird's into the Jell-O name, and mass-marketed the varieties of flavored instant custards here in the US, where they were known almost immediately, as puddings. After a while, the need to put other things into your Jell-O faded, but the name, "Pudding," stuck.