I know that what the UK calls crisps are called chips in Yank-land and (Frenched) fries are pretty much the same, but if I wanted the voluminous British chips rather than the tiny American fries, for what should I ask in America?
Potato wedges?
Steak fries should match what you’re used to.
Thick fries?
Of course, there are chippies around, so it’s not impossible to find British-style fries. If you go into a fish and chips place, they’ll know what you’re talking about.
Can you link to a photo showing what you’re picturing?
There are a couple of different types of potato that can wind up going under that name in the U.S., and I’ve regularly seen both types sold under that name.
One (which I suspect is the style that the OP is looking for) is essentially just a bigger / wider version of the “shoestring” style French fry. The other is more of a fried potato wedge (with the skin on).
Yep. Here’s an example of fish & chips from a local Irish-style chipper. I miss them; the nearest one is a Starbucks now.
They’re all French fries (or “fries” or, rarely, “potato fries,” all from “French-fried potatoes”). We don’t distinguish between fat and skinny fries. Rarely if ever will you have to distinguish between them, because any one establishment will have only one style of fries available.
Fries can be enormously variable from one place to another, but only rarely does a restaurant describe them with specificity, unless it’s something that they emphasize as something they specialize in. So you might very occasionally see “shoestring fries” or “waffle-cut fries” or “potato wedges,” but it’s rare you will find the need to specify what style of fries you want.
Yup. Must resist…
Resistance failing… 
Yeah, I’d still call those “fries.”
Maybe “steak fries,” but they’re still fries.
in the frozen food section of American supermarkets you can find a variety of different French fries. The ones you are describing are called ‘steak fries’
It’s clear you have a very specific idea of what a “chip” is, but, after having spent half a year in the UK, it’s close enough to what I would call a “steak fry” here (although, as pointed out above, it can refer to a couple of similar potato products.) To me, a chip is just a fatter (and often stubbier) fry. As to what to ask for? There’s nothing specific. Places sell their French fries as they sell them. Steak fry, to me, would be the closest, or you can say British/Irish/whatever-style chips/fries and see where that gets you.
Indeed – Ore-Ida (the biggest manufacturer of frozen fries in the U.S.) makes “Golden Steak Fries,” which look to be pretty close, and those are probably the norm for private-label / store brand ones, too.
Once we’re talking about restaurants, however, “steak fries” might wind up being the potato wedge style, too.
Absolutely so. On occasion, you’ll find a restaurant that offers a couple of different styles of fries to choose from, but those are the exception to the rule, and if there’s a choice to be had, it’s more common that it’ll be regular fries vs. sweet potato fries, rather than different cuts of regular fries.
Or curly vs regular fries (with curly fries usually – actually, always in my experience now that I think about – being seasoned.)
“Curly fries. Hand-picked. Huh-huh! Huh-huh!” 
Did you hear about the French gangster Pommes Frites?
His friends call him “Chips.” 
When they are that thick, I do prefer calling them "steak fries’. Actually, I don’t like fries that thick, the center is usually mealy. To each their own, of course.
This is the definitive answer. Most places just have their own style of fried potatoes, which they call “fries.” There used to be a place near me that offered a good appetizer that was your choice of 3 styles (which were really only selections from the entire McCain frozen lineup, so probably nothing like chips), but most places have one type of dinner/lunch fried potato side and one for breakfast (hash browns, potato cakes, home fries, hash rounds, silver dollar potatoes), if applicable.
I am not a fan of the shoestring or crinkles that remind me of school cafeterias. McDonald’s seems to be the exception of a tasty fry that consists of very little potato, IMHO. I grew up with Thrasher’s boardwalk style fries (much like I grew up with “New York” buy a slice and walk the boards pizza), so that’s pretty much my standard. As others have mentioned, some “authentic” fish and chips spots might be the best bet.
Steak fries or country fries - not to be confused with country fried potatoes, which is the same as country hash browns.
I can attest that Eamonn’s Dublin Chipper makes tasty food.
But don’t you dare crowd that place up.