How to update Windows 10 that's nearing the end of support but says it is up to date

The Windows Update screen shows the following messages and options.

1. A message "You're up to date. Last checked: Today, 11:59"
2. A button "Check for updates".
3. A message "You're currently running a version of Windows that's nearing the end of support. We recommend you update to the most recent version of Windows 10 now to get the latest features and security improvements."
4. A link "Learn more"

The messages 1 and 3 look contradictory. The PC is both "Up to date" and Microsfot "recommends to update to the most recent version of Windows 10 now to get the latest features and security improvements. "

My question is how to apply the recommendation mentioned in message 3? Is there a change that there is a problem in the Windows Update system? Or are the messages just contradictory and the PC is updated.

I describe what I see textually and also add a screenshot below for completeness.

The messages 1 and 3 look contradictory.

They're not since the first message tells you that you're up to date on the version you're currently running while the third mention the Windows version you're running is ending its support (Windows lifecycle fact sheet):

My question is how to apply the recommendation mentioned in message 3?

Go to the Windows 10 page and on the top you will find a button that says Update now. That will download Windows 10 Upgrade which you then can update your Windows 10 using it.

Is there a change that there is a problem in the Windows Update system?

I'm not really sure but with PC's I have around with Windows 10, none seems to retrieve feature updates via Windows Update despite my installations are older than 31 days old and I don't have "defer updates" option enabled (as suggested here).

The messages are not contradictory. You need to upgrade (it is telling you that), but you will still get security updates for a while.

If it does not update when you click the Update Button, run the Windows 10 Repair from the the Microsoft Media Creation link and upgrade that way. Watch the process and select option (default) that keeps data and applications both.

https://www.microsoft.com/en-us/software-download/windows10

Windows 10 is running, so click on the Download button (not Upgrade Button, select Save. Create a USB Windows Installation key and then run Setup on the USB Key. This will launch the Repair and proceed normally.