You're on a hiding to nothing saying the pick up wasn't invented in the USA.
Like it or lump it this is an American site. From what I've seen the vast majority of members, readership and subject matter are American.
'Tray' wouldn't make much sense to the readership, IMO.
'Ute' invokes a style of flatbed/flatdeck as seen in Australia. If there is missuse or confusion of the term I attribute the root of that to some importing into or selling that stye of product in the USA. eg UTE LTD.
If in doubt a picture should help.
Idk. I’ve only ever seen what the OP is referring to called a tray, even in America. But I was raised in the heartland and trays on pickups (with the bed removed) were common place on ranches and farms. Most of the domestic dealers stock their HD trucks without beds fitted specifically to allow for boxes and trays to be fitted immediately.
I’ve also understood Ute to mean something that is basically a sedan with a small truck bed in the back, like posted earlier. The Ford Falcon (from the 1950s), Ford Ranchero, and Chevy El Camino are all Utes. I think most Americans with the slightest interest in cars will know the old classics, and anyone interested in modern cars will know the Holden and others.
I take a canopy, topper, or camper shell to be something that goes over a pickup bed and provides a high capacity, enclosed area.
OP is asking about a tray, but let’s not split hairs over semantics and give them some proper feedback.