The password you entered to protect your iPhone backup could not be set.

I am grateful for your attempt to help me. However, as a true blue, long time Mac/Apple everything person since 2007, you can be sure that I did read and follow the instructions (precisely those which your link provided) carefully, and followed every single step each time that I tried and failed to get it to work. Every step. EVERY step. There is one thing that the instructions omit.

The instructions say this: "After you confirm your password, your backup will start and IMMEDIATELY overwrite and encrypt your previous backups." [capital letters mine] That is incorrect. The backup would not start, not only "IMMEDIATELY" but never, not once, because it kept insisting that I enter my iPhone password on my iPhone (or at least it seemed that is what it wanted me to do as a prerequisite before beginning to prepare the backup, so it could be assured that the phone was safe, or that I owned it, or ..... who knows?). In other words, there is a step that as, being one human, the only human in this room, I could not carry out-- I could not do both things simultaneously --eg (a) type the password for encryption onto my Mac using my Mac keyboard AND (b) type the iPhone's password onto the screen of the phone. I could not perform both tasks simultaneously to the satisfaction of the maniacal and demanding robot that runs this system. I will try to send you a screenshot of each step when I get the chance, but in the meantime I feel the need to defend myself !!

I have a high IQ, I am wildly over educated, and I love everything Apple. I follow instructions carefully, it's built in to my life (retired lawyer) and in this case I followed the instructions that were given to me over and over and over again !! So alas your kind provision of the steps I had already diligently followed did not help me. I am open to any other ideas or suggestions you may have. THANK YOU.