Thank you, that helped!
When I first tried your suggestion, My Photos was not displayed as an option at all, but you got me on the path to knowing it's possible. A couple of extra steps turned out to be needed before the album appeared. Here are those steps for others who might be reading this and looking for a solution:
To make My Photos appear under Settings > General > Screen Saver > Type:
- Go into Settings > Users and Accounts > (my name) > iCloud Photos and turn that ON
The only album that appears under My Photos is Favorites. No other albums appear (they might appear after about an hour of waiting - see note below). If that isn't the album you want displayed, turn on shared albums:
- Go into Settings > Users and Accounts > (my name) > Shared Albums and turned that ON too.
Activity, Favorites, and Shared Albums should now appear under My Photos. However, My Albums might not appear. To create a shared album:
- On a MacBook, open Photos.app, create a new album under Shared Albums (not My Albums), and call it something like "Apple TV screen saver" then drag the images you want into that album.
It takes a few minutes for the album to appear on the Apple TV under My Photos. Once it appears, select that album and the images will be displayed as the screen saver.
The key here is to turn on iCloud Photos and Shared Albums (which are off by default, even if you've signed into iCloud on your Apple TV with your Apple ID) and then put the screen saver images into a Shared Album.
For some reason, the Shared Album has to be created on a Mac running Photos.app. It does not work to go to icloud.com in a browser and create an album in there, because that is a My Album not a Shared Album. If you don't have a Mac, you can do it from your iPhone by going into the Photos app and creating a shared album in there. It will ask who you want to share it with, just leave it blank.
Note: After I created a shared album on my iPhone, my entire iCloud photo library (including all of the albums under My Photos and every photo I have ever taken) suddenly appeared on the Apple TV. This isn't necessarily what I want either, and I don't know whether it had anything to do with creating a shared album or just a coincidence that took its time, but at least it's working. Thank you for your help!