With macOS Sierra did you download the InstallOS.dmg from the link you posted.
you then open that to get the InstallOS.pkg, you open that and you will see an
installation window.
This does not install macOS Sierra after all the information in the window says it will
only take up 14MBs of disk space and only took 5 seconds to complete..
No installation of a full OS only takes up 14MBs and 5 seconds to install.
What is actually happening is it is converting the InstallOS.pkg to the Install macOS Sierra.app
which you will find in your Applications folder.
Double-click on this to install macOS Sierra.
If you want to install macOS High Sierra it is very important that you use the Safari browser
to 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.
You will find that your MBA from 2013 can install all available OS's, if you want to
install Mojave or Catalina I would recommend you install macOS High Sierra first.
Instructions for installing the latest macOS Big Sur are below.
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'.
If you want to install Big Sur, open Safari and click on the link below.
How to upgrade to macOS Big Sur - Apple Support
If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.
Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,
some updates may be free, some may be paid. Use this app to discover which apps you have
are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see
if they are offering 64bit updates.