My emphases.
Bread pudding and flan, etc. are custards.
As for Yorkshire pudding, we just don’t have an American name for it. So it falls within the British English (BE) usage.
Then again, there’s an egg-based dish I like to make for breakfast on weekends, which apparently is similar to a very large Yorkshire pudding, and which is called in the US a “German pancake” or a “Dutch baby.” So apparently something called pudding in BE can be called a “pancake” or a “baby” in the US. 
It’s probably one of those false anachronisms they like to use for Mr. Burns, but there’s a kernel of truth to it, in that there are basically two types of ice cream- the kind made from milk, cream, and sugar, and the kind that’s basically a thin custard made with eggs that’s then frozen in an ice cream maker. Some places call it “frozen custard” to differentiate themselves, but it’s still ice cream in the same broad sense that gelato and sherbet are forms of ice cream.
So the idea of “frozen pudding wagon” is probably playing off the custard style of ice cream and making it sound quaint and goofy.
The bread pudding I know of is more like a fruit cake where bread is used instead of flour - there is a sort of custard involved in that the wet part of the mix contains milk and eggs, but I wouldn’t describe the end result as anything like a custard.
Are you referring to something else?
Corn pudding is relatively popular in the South, and is a staple side dish at many barbecue joints.
That’s my experience with ‘bread pudding.’ Similar to fruit cake, but without the goodies. Southeastern USA.
Around here (Ohio and Pennsylvania), bread pudding is just taking a bunch of old, probably-stale bread, and smushing it together with milk and eggs as a binder and some sugar, and then baking it. It might also have raisins.
I always thought bread pudding and rice pudding were precisely analogous: a cooked starch saturated with a custard and cooked until the custard sets (with further thickening contributed by starch).
The starch food settles out in the bottom yielding a custard layer on top, if you use enough custard.
Wait, you mean corn pudding is an actual thing, and not just something made up for a silly song in Schmigadoon?
You take that back! (I love that song.)
Bread pudding (as opposed to bread and butter pudding) is something that you can cut into squares and pick up. Whilst that’s technically possible with rice pudding, it’s not the normal presentation (in the UK versions I have encountered I should say)
My bread pudding is cubed stale bread with egg, milk, cinnamon, nutmeg, poured over and baked.
My rice pudding is pretty much the same, but with rice instead of bread, heated on the stove until thickened, then cooled.
Raisins/dates/figs can be added to either.
Wow. I would never describe it that way, but I can’t tell if that’s based on minor differences in how it’s prepared–like a “drier” version–or if it’s a bigger difference.
It makes me wonder about fruitcake too now.
I would say it’s putting stale bread in a dish, making a custard base, and pouring the custard over the stale bread. The bread absorbs the liquid and becomes totally soft. When the custard is set by baking it, the whole thing has a custard texture. Sometimes a bit stiffer, but definitely a custard.
I can’t fully tell, but I think my version might be more like bread and butter pudding? Though I’ve never seen or heard of anyone buttering the bread. I would not want to pick it up, though it might technically be possible. It’s not a cake texture. And the liquid is egg based, not milk based.
Brit to U.S. translator:
Brit U.S. ----- ----- Pudding Any number of dessert type treats Custard Pudding Biscuit Cookie Scone Biscuit Chip French Fry Crisp Chip
Separated by a common language, indeed.
These are my recipes, too. And both are really custards. And i cut slices of either.
I’ve seen a soft “rice pudding” that’s more like tapioca pudding thickened with rice. It’s actually a pudding (in American) and i don’t care for it, so i never make it. Tapioca pudding is also a pudding. “Black pudding” is a sausage, and sometimes it’s called “blood sausage”, which is a better name for it in the US. “Yorkshire pudding” is a British dish like popovers or a Dutch baby, but made with beef tallow instead of butter. It’s not a pudding, it just has “pudding” in the name.
I guess that means that a pudding, in my parlance, is a sweet dessert that’s soft and is primarily milk thickened with a starch, plus something to flavor it. By default i assume it’s chocolate.
Yeah, bread and butter pudding is normally small slices of bread, buttered, arranged in a dish with raisins scattered over it, then an egg custard mix poured over to saturated the bread. Sugar and spices, then baked to set it - texture is wobbly with a crunchy top.
Bread pudding is very similar ingredients but different proportions and the bread is torn into small pieces.
Oh, and “plum pudding” is a food in British books, old songs, or maybe in operas. I have no idea what it is, but i bet it’s not actually a pudding. I vaguely recall being told it’s not actually plums, too, but maybe I’m getting confused.
You mean that you’d pick it up and eat it with your hands, as opposed to a utensil? That doesn’t match my experience of bread pudding.
The way I understand it, “pudding” in UK usage also refers to the course where those treats are served, which makes it essentially synonymous with “dessert.” Thus, Brits say what sounds to my American ears like “We’re having [food item] for [completely different food item].”
“We’re having cake for pudding.”
“We’re having scones for tea.”
Sounds about as bonkers as…
“We’re having brownies for pie.”
“We’re having pork chops for sausage.”
“We’re having potato chips for beer.”
It sounds like you might be describing what is known to me as bread and butter pudding
Could be, but I don’t remember ever encountering the term “bread and butter pudding.”
In my experience, bread pudding (called that) is fairly common in places like cafeterias and buffet restaurants, because it’s a good way to use up leftover bread that’s gone a bit stale. It’s served in a big tray, and you can dish up a bowlful with a serving spoon.