The Google Photos interface is very minimal even for humans, let alone applications. The best way I found to get a list of album names and URLs is to parse the page source using the browser console.

The list of all albums

Executed on the page https://photos.google.com/albums, this script returns a string that can be copy-pasted to a Google Sheet, creating a table of names and URLs.

var links = document.getElementsByTagName('A'); 
var s = ''; 
for (var i = 0; i< links.length; i++) {
  if (/\b(album|share)\b/.test(links[i].href)) {
    var albumName = links[i].children[3].children[0].innerText;
    s = s + albumName + '\t' + links[i].href + '\n';
  }
}

The list of shared albums

Executed on the page https://photos.google.com/shared, this script returns the list of album names and URLs.

var links = document.getElementsByTagName('A'); 
var s = ''; 
for (var i = 0; i< links.length; i++) {
  if (/\b(album|share)\b/.test(links[i].href)) {
    s = s + links[i].innerText + '\t' + links[i].href + '\n';
  }
}

The URLs are in the long format: https://photos.google.com/share/...?key=..., not in the short format https://goo.gl/photos/... that you would get by clicking the share button. But they are functionally equivalent: the latter redirects to the former after going through Google's URL shortening service goo.gl. If length is a concern, you can shorten them yourself within Google Sheets by using an Apps Script for generating goo.gl URLs.

There is no right combination of settings.

The Google Photos upload checkbox (in the Backup and Sync preferences) is a blanket toggle: it affects all the selected local backup/sync folders. So if you want to backup Drive-only files, you need to leave it unchecked and continue manually uploading your images to the Photos app.

As of yet, the B&S client does not allow fine-grained control of what gets uploaded into Photos. Also for backups, be reminded that Photos has limitations on file types and min/max sizes.

Workaround

As a workaround for periodic backups to Photos, you can try the following:

1. (Make sure you don't have Drive and Photos synced with any of the 2 separate settings you mentioned)
2. Open Backup & Sync Preferences → [your device]

3. For "Removing item", select "Don't remove items everywhere".

   

4. Select only the folder(s) you want to back up to Photos

5. Select the "Upload newly added photos [...]" checkbox under the Google Photos Heading. Wait for everything to back up.
6. Uncheck the checkbox under Google Photos again.
7. Uncheck your images folder and re-check your Drive-only folders.

As a side-effect to using B&S, you also retain the images in your Google Drive. But they are linked invisibly to Photos, so:

• they don't eat drive quota (if HQ setting was chosen), and
• deleting them will also remove them from your Photos library.