We call them hatchback also. I guess most of them are compact size but saying compact would only identify size and not type. Could be hatchback/sedan/coupe although there aren't many coupes left anymore.
That's easy enough, the S's go together, saloon/sedan. I may remember that even past lunch.
A hatchback might be called a liftback. I think it is something a marketing department came up with for a hatchback with a less steep and larger hatch. Aside from certain models, I don't think the public uses it much. Like the 4 door coupe.
You can have the shape of a saloon, but if the rear screen also rises with the trunk lid, then it's referred to as a fastback, somehow makes it sound more sporty and upmarket.
The first official liftback I recall was the Prius. When Toyota was distinguishing it from the c and v. Or maybe it was the community, as Toyota does call it a sedan too. If the hatch is like what you expect on a wagon, it can't be a liftback.
The classic Mustang fastback has a trunk, though the rear window is sloped enough you'd think it had a hatch.
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