Box, enclosed, cargo, dry van — four words for overlapping things. What each one means to an exporter, and which one describes the trailer you actually need.
Four words describe overlapping things in this corner of the market, and which one you use changes what you get quoted.
Dry van is the international freight term for a sealed, non-refrigerated semi-trailer body. It is what this vehicle is.
Box describes the shape. Add a heavy-vehicle qualifier — a box semi trailer, an HGV box trailer — and it means the same vehicle. On its own it usually does not.
Enclosed describes what the body does rather than what it is. Useful as an adjective, useless as a category.
Cargo describes the contents. It narrows nothing at all.
None of these is wrong English. The problem is that three of them are used for something else entirely in the largest English-speaking market.

In North America, an unqualified box trailer is overwhelmingly a small enclosed towable — 6 × 12 ft, 8.5 × 30 ft — bought for moving house, hauling motorcycles or fitting out as a workshop. Search for it and you get consumer trailers on a ball hitch, not freight equipment on a fifth wheel.
The same is true of an unqualified enclosed trailer and an unqualified cargo trailer. These are real products with real markets; they are simply not this product.
The consequence for a buyer is practical rather than academic. Sending an enquiry for "enclosed trailers" to a supplier list assembled from those terms gets you quotations for vehicles a fifth of the size, and wastes a week finding out.
Add the qualifier and the ambiguity disappears. Box semi trailer and HGV box trailer land squarely on heavy freight equipment.
An export dry van, as the term is used outside North America, is a sealed semi-trailer body sized to the combination length limits of the market it works in — typically around 12 to 13.6 m of body rather than the 16 m of a North American 53-footer.
Ours is 12.9 m internal, 2.45 m wide, 1.8 m high, 56.9 m³, on three axles, with ABS and axle brakes fitted as standard. Sealed for the whole run, because the point of the format is cargo that must not get wet, must not blow off and must not be reachable at a border queue.
Use dry van trailer if you want the enquiry to land on freight equipment first time. Use box semi trailer or HGV box trailer if that is the phrase your organisation uses internally — both are unambiguous once the qualifier is there.
Avoid unqualified box trailer, enclosed trailer and cargo trailer in a supplier search, not because they are wrong but because they will fill your results with a different product.