I'm back. From looking at your two graphics, I obviously have no concept in using Smart Albums or referencing in Photos. I assume these dialogues can be found in preferences? And can just any album be made into a Smart Album?
You can create smart albums from the file menu, the images I posted were to provide the settings for you, the purpose of creating these smart albums is for diagnostics only, you can delete them afterwards,
Yes, I did sign back in again on iCloud via the Apple TV. And I have a list of albums (i.e., Photos, Selfies, Macworld 2009, Snowstorm 2010, ...). Do these need to be turned into Smart Albums?
Are those the albums that you see on the TV in the Photos app. The reason I ask is because this seems odd. The first two albums (Photos and Selfies) shouldn't ordinarily be there, they are albums that are generated by the Photos app on your other devices and part of the iCloud photo library which isn't shown on the Apple TV. Possible explanations for you seeing these albums is that you have created shared albums and given them these names or that you are looking in the Computers app on the Apple TV instead of the Photos App.
What do you mean by 'Activity'? I see that listed on the AppleTV and my MBA with no pix, but on none of my devices.
Activity is an auto generated iCloud Shared album (iCloud sharing is a wholly different and separate set of photos to the iCloud photo library-although the same photos can be in each), it's a sort of smart album that keeps a track chronologically of activity in iCloud sharing, it shows the last photos you (or others) have added to iCloud photo sharing or/and any comments that have been added to them).
On your devices if you tap the 'shared' option at the bottom of the screen, do you not see an 'Activity' album.
If you have no photos in 'Activity' this suggests you haven't made any shared albums, which means that you shouldn't see anything other than an empty Activity album and a Photo Stream album under the 'Photos' app on the Apple TV, which is why this is so very confusing.
Strange. One of my folders is Called 'AbilitiesExpo 2012' and should contain pix from only that event. However, it contains selections of pix from all of my albums (sort of a truncated version of 'All Photos.'
Folders aren't albums, but they can contain multiple albums, so what you describe makes sense to me. If you look at a folder you will see photos from all of the albums within it.
I'm becoming more and more confused, obviously.
Is it possible that my MBA is the problem? From time-to-time it does flaky things.
OK, lets see if I can't explain the Photos structure a bit better for you. The first thing I suggest if you haven't done it already is to look at the Photos App on the computer in Sidebar view, you can do this from the 'View' menu and choosing 'Show Sidebar'. Below is my sidebar.


There are 2 images simply because I couldn't get it all in one screenshot. From the top...
Photos
These are all the photos in your library (iCloud Photo library, if you have it turned on), Photos in this section are listed chronologically according to the date they were taken (unless you have modified that date. You can condense this view and these views are called moments, collections and years)
Import
If you connect a photo device, such a s a camera or phone it will appear here and let you import from it. If you hover your mouse over this heading you can hide/unhide the devices that are connected.
Shared
This is another heading that you can hover over and hide/unhide the contents of. Shared albums are albums that you create so you can share them with other people, you can also create them to share with nobody, but so that you can see them on your Apple TV. This is iCloud Photo sharing (not iCloud Photo library) you will see these albums under the 'Shared' heading on a mobile device.
Albums
This is another heading that you can hover over and hide/unhide the contents of. Albums is a mixture of smart albums and manual albums. Some smart albums are created by the app itself, such as Selfies, Panoramas, Videos others are created by the user such as 'Upload failures' and 'ASC Photos' in my case. Manual albums are created by the user and can be listed under Albums in their own right like the album 'ASC Photo Album' or '0000' in my case or they can be contained in folders such as 'Prints' and 'Events' in my case. You can click on the triangle at the side of a folder to see what is inside it.
Projects
These are calendars, slideshows etc that the user has created.
Photo Stream
You can't see the photo stream album in the Photos app if you have iCloud photo library turned on as well as photo stream in the Photos preferences, however if you do have it enabled it still functions in the back ground. The photo stream album can be seen by the Photos app on the Apple TV.
On the subject of the MBA, I'm not sure at this moment in time, I suspect a lot of the problem might be because you aren't entirely sure what you should see where.