What part of a cabinet is the "carcass"?

I agree with keshlam, but I want to share other info on the other aspects of your question.

The carcass is the main framework, or body, of the object. It is what remains after you strip all coverings, doors, front-face panels, frames for these panels, and (non-structural) shelves. It is the framework of a structure, and is generally not seen.

A "frame" would be part of the carcase if the word was employed in the general sense of "chassis" or the "structure". The word "frame" can also describe a particular form of construction/joinery, as explained in the next quote.

The European cabinet example you have shows a frameless carcase (a.k.a. "full access" cabinet) -- the door panel hinges are attached onto the carcase directly. The American one has a face-frame -- door hinges are bolted on a face trim which is itself attached to the carcase. The term "frameless carcase" would suggest that frames could be part of a carcase, but it would be more accurate to say that they are rather attached to it, like in American example.

Fine woodworking makes an even more clear-cut distinction between frames and carcases:

Furniture construction is broken down into two main categories: frame and carcase. In frame construction, relatively narrow boards are joined—usually with a mortise and tenon joint—as in a chair or table base, or in a frame and panel door. (See Fine Woodworking, Summer 1976.) In carcase construction, boards are joined end to end using dovetails, tongue-and-groove joints and the like, as in a drawer or hutch.

Where do panels fit?

That quote above doesn't mention panels as being a category. The word "panel" can ambiguously mean a series of (one or more) boards side by side forming a flat surface. Or it can mean a decorative pane. For instance, each of the constituent faces of the box that constitutes a carcase is itself a wood panel. So whether the back panel of a kitchen cabinet constitutes integral part of the carcase would depend on its function being structural or not, like keshlam mentions.

In other contexts:

The form "carcase" (plural "carcases") is closer to Middle English spellings. The form "carcass" (plural "carcasses") is cognate with the French "carcasse". "Carcase" may be more common in varieties of British English than it is in US English. (1)

I don't recall encountering the spelling "carcase" outside of woodworking texts, e.g. to designate the remains of a burnt building, or of a turkey dinner, or cadaver. But, apparently usages in older texts are found for both spellings in all the possible different definitions and terminologies. So neither is specific to woodworking, nor does there seem to be definitions covered by one spelling, but not by the other -- with the exception perhaps of "carcass" as an early incendiary projectile device.

As far as other uses of the word go, within the realm of woodworking, aside from (1) the carcase of a cabinet, and (2) carcase construction mode, I haven't found anything interesting.

Popularity of Carcass vs Carcase, in English texts over time from Google-N gram, all definitions combined: