How do I show Finder menu-bar icon names?

The application name is buried within the binary of the menu bar extra item, so AppleScript cannot get at it. You can get the description of the menu bar extra item, if the developer chose to give it a name. Malwarebytes did not as you can see from this unsorted, AppleScript output based on the following menu bar extras:

tell application "System Events"

get description of every menu bar item of every menu bar of process "SystemUIServer"

end tell

Result:

{{"Notification Center", "Siri", "VPN", "Wi-Fi, four of four bars, with bifrost 5GHz.", "bluetooth", "time machine", "text input", "Clock", "User", "volume 44%"}}

PS: Right-click on the Finder menu bar on either end of it does not present a viewing choice such as described by others. I am using macOS High Sierra 10.13.6 (17G65).

What should I do to 'unwonk' it?

Nothing, all of the previous replies were giving you guidance for the Toolbar, not the menu bar. There is nothing that shows the names of the icons in the status menu.

I couldn't find an article that describes them, short of the articles describing the particular item (e.g. Time Machine).

Also, you could have icons put their by third-party software.

Select Customize Toolbar from the Finder's View menu. Select the option from the dropdown list in the bottom-left corner of the window. Click on the Done button.

It works for me in High Sierra.

Works for me just fine.

Not so on my system. Try the way I posted.

Doesn't just clicking on an icon bring up a menu until you click it again?

Right-click anywhere on the finder menubar and choose, "Icon & Text".

Sounds like your Finder preferences may be wonky. It should work, either way.

If you don't have an open Finder window then there is no toolbar.