The App Store Updates page only informs you of updates available for the OS you are running.
Installing a newer OS is called an upgrade.
For macOS High Sierra,
Open Safari and click on this link,
How to get old versions of macOS – Apple Support
Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.
Then go to the Download OS section and click on the OS you want to install.
At the time of writing this the links contained in the Download OS section
for macOS High Sierra, Mojave and Catalina are not working, Apple have been informed
and we are waiting for them to restore their functionality, so maybe try again later in the day.
Before upgrading make a full back up of your Mac.
Check that third party apps you use have updates available for the new OS.
https://roaringapps.com
Check that your printers/ scanners etc have updated drivers and software available for the new OS.
Connect your mac to your router via cable rather than rely on WiFi.
Do not use your mac while it is downloading
and do not use any other devices that share the same internet connection.
If the App Store says ‘Open’ then it is detecting a previous download of the install app
go to the Applications folder locate Install macOS ******.app.
Move it to the Trash, empty the Trash.
Restart, click on the link again, the App Store should say ‘Get' or 'Download'.