Geek_Girl and AlexT,As far as I can see, all my permissions were correct.
Although Photodeluxe was already off, I eventually deleted all the photos from my albums and members albums and started from scratch, experimenting as I went.
Using the Dolphin Photo module only:
When I loaded a photo into the default album "Nickname's Photos" it immediately became the profile photo and avatar. As soon as I added another, the new photo became the profile photo and avatar. Obviously Nickname's Photos should be called Nickname's Profile Photos to avoid confusion. Clicking the Set as Avatar button did switch photos, but any new photos added to the album would override it. I guess that's logical and all new photos should be prospective avatars.
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Please read the attachment before commenting.
I've now created three albums and I'll ask my members to do the same. Profile Photos (I've renamed it in Admin/Photos Module), General Photos, Friends Photos (both set Members Only) . It would be great if Dolphin could create these three albums from the outset. It would save so much confusion.
By selecting a profile photo in the default (Profile) Photos album, it now appears in all the right places.
When I place additional photos in the other albums, they're now appearing in the correct places also.
Please note: Although I'd removed PhotoDeluxe, the original problem persisted until I eventually deleted all the photos and replaced them.
I'm now about to work on photo deluxe to see if it compounded the problem.
The attachment isn't so much to do with this problem, but more about why the incorrectly named "Nicknames Photos" Album confused me and possibly many others.