Message: Your iPhone cannot be backed up… etc. while I have iCloud Backup turned off

Check in settings under iCloud and make sure that nothing else is on attempting to backup or sync to iCloud (like Photos or all of your apps). Even though you don’t have backups on you may have one or more of your apps syncing to it. This may not be the case but it won’t hurt to check to be sure. Otherwise you may need to contact Apple Support directly for further assistance

tupppi Said:

"Message: Thanks so much for your reply. But I’m not sure I understand some of it… [...]Roughly 97.4% of the iCloud storage bar on the phone is grey but it’s not labelled ‘Other‘ or anything else.[...]"

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Thank you for the screenshot.

You Need to Upgrade to an iCloud+ Plan:

"Other" is for Internal Storage:

"Other" (now called "System Data" in more recent iOS versions) applies only to internal storage (that on your iPhone), and is never stored in iCloud storage. Looking at the internal storage, what do you see there(Go to: Settings > General > iPhone Storage)?

So, Upgrade to an iCloud+ Plan:

So, all you have is 5GB of iCloud storage (which Apple gives you as a user). So, upgrade your storage plan, and you would not get this error message. Go here: iCloud+ Plans and Pricing - Apple Support

tupppi Said:

"Message: The iPhone has 45.2Gb free out of 60Gb. Why do I need to buy more storage? What is this extra storage for? What needs to be stored? I'm sorry, but I really don't understand what indicates that I don't have enough storage.”

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Thank you for the screenshot.

You only have a 5GB plan. That 18GB would no upload. So, try clearing all of that grey bar, seeing if you then let less than 5GB: How to Clear "Other" in your iPhone's Storage - Use Backup - User Tip

tupppi Said:

"Message: Your iPhone cannot be backed up… etc. while I have iCloud Backup turned off: For many months I‘ve been getting the message that the phone can't be backed up because I don't have enough storage despite the fact that iCloud Backup is off. [...]"

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Clear iPhone and iCloud Space:

A. Clearing Space of an iPhone:

Clear Space Using my User Tips:

See if you need to clear lots of space. System resources may be using up too much services, making it overheat. Internal storage cannot be upgraded. It is what it is. So, the maximum internal storage capacity never increases, even if you upgraded your iCloud plan. Learn More here: What's the Difference Between Device Storage and iCloud Storage? Apple Support

So, what to do...

Clear Space, Using my User Tips:

• How to Clear Space from your Mobile Device, and What to Clear: - User Tip
• How to Clear "Other" in your iPhone's Storage - Use Backup - User Tip

B. Cleaning Space of iPhone:

One unique method...

Download All of iCloud:

One thought would be to download all of iCloud to an external drive. Once downloaded, verify your can access the data. Once verified, trash what is in iCloud. As for seeing through the download, do so in Finder on a Mac. You would download it to an external drive, and then open the drive in Finder. Search through it using the search bar on the top-right corner of the Finder window. Once downloaded, trash what is in there. Use my user tip: Getting a Quick and Direct "iCloud Drive-to-External Hard Drive" Download - User Tip

The OPs screenshot indicates they have plenty of space and no need to get iCloud+. They appear to have no desire to backup to iCloud and they don’t have an abnormal storage allocation.