Each Space can use the default Desktop, or have it's own Desktop, which can be fully customized.
It is confusing because of Apple's inconsistent use of terminology, and it is not immediately clear what the relationship is among Mission Control, Spaces, and Desktops
Here's my current understand of these terms, subject to further edification:
Mission Control
Management tool for Spaces
Allows you to see what's in a Space, Change Spaces, Move apps to a Space
Space
A customized collection of desktop and apps
Desktop
The screen object you actually see
Has properties that can be customized from System > Desktop
• Can have zero or more icons/objects shown on it.
When you change it, you are changing the Desktop for the current Space
To belabor a point, my OP showed how Apple uses the terms inconsistently, somewhat interchangeably, and sometimes in the same phrase. I wasn’t trying to sort out the truth, I was trying to point out that you can’t find the truth from Apple’s usage, including its documentation.
What shows icons/objects is neither a space, or a space-type desktop, nor the Finder. The icons for objects and folders are there no matter what application is current and whether or not the Finder is hidden. (I use Desktop Curtain to hide all that clutter.) They are on what has from the beginning of (Mac) time been referred to as a desktop — that was probably the primary metaphors on which the Mac GUI was based. My reference to the Finder desktop was a lazy inaccuracy — what I meant is the ancient desktop that shows icons which the Finder manipulates, but isn't the Finder itself.
What's this? In the Mission Control world spaces and desktops, whatever they mean, have a 1-1 relationship.
One way to come at this is to consider the Desktop & Screen Saver Preference Pane, where you can set the color or picture of a "Desktop". Each space-desktop has its own color or picture as its background. Hmmm -- what gets the color or picture is, I think, yet another use of the term "Desktop". That is, there is a background called the "Desktop" on top of which are file, disk, and folder icons, on top of which are application windows. When you switch to a Desktop/Space with Mission Control or its keyboard shortcuts, you end up with a possibly different background color or space, the same set of "desktop" icons, and 0 or more application windows, some of which may appear on all desktop-spaces, and some of which may be assigned to the specific space-desktop you switched to.
I hope that confuses things further :-). It's an interesting research project.
OS X El Capitan: Work in multiple spaces
If your desktop becomes cluttered with open app windows, you can create additional desktops, called spaces, to organize the windows. When you work in a space, you see only the windows that are in that space.
You use Mission Control to show the Spaces bar, where thumbnails represent your spaces and apps you’re using in full screen or Split View.
As of this moment, I believe that "desktop" mostly refers to the original desktop, which is actually a Finder object.
"Spaces" are "virtual desktops", that can contain windows of different apps (but all windows of a specific app are on the same "Space"), and have a different background color/image. But all "Spaces" share the same items in your ~/Desktop folder, which are displayed on all Spaces.
This issue is confusion because Apple has been changing definitions and terminology with almost ever new OS X release since 10.8 (maybe earlier).
Maybe the easiest way to keep clear on this, is to always think of the "Desktop" as a Finder folder.
“you can create additional desktops, called spaces, to organize the windows” in my original post was a quote from Apple’s help document OS X El Capitan: Work in multiple spaces So in at least one place Apple says desktops are space. I don’t know why the writer of that document didn’t see the weirdness of stating that equivalence.
