We do have distinctions, but most eateries only have one kind of fry. So you can ask for steak fries, but they may only have shoestrings.
My experience with Fish&Chips here in New England, at my own restaurant and others, is that 3/8" to 1/2" square cut fries are served with the fish. A steak fry would be rectangular, at least twice that size, and that’s what I recall eating in England.
If they’re going to be counting tater tots and cottage fries as “kinds of French fries”, then they really ought to have crisps/chips, too. Probably several types of them.
Looking at that chart, I’ll note that “smileys” (the last entry on their list) are typically not sliced-and-fried potatoes. The biggest manufacturer of those is McCain, with their “Smiles” product, and they’re actually mashed potatoes with a crispy exterior.
I don’t know if you’re just asking for edification, or you’re planning on a trip to the USA in the near future, but be aware it is vanishingly unlikely that asking for steak fries will work unless that’s something the restaurant already serves. So don’t jump right to the insult tip because the kitchen only makes regular fries.