Hi, Thanks, I do use Firefox and Chrome as well, but wanted to at least have the latest version of Safari, version 13.1.2, which IS available for this version of High Sierra's last published version, 10.13.6, but only available under special circumstances.
The last version of Safari 13.1.2 was published on 15th July 2020. It isn't included in any of the security updates for High Sierra 10.13.6, available here: Apple - Support - Downloads
And reinstalling High Sierra using Recovery mode (restart holding down command R) doesn't give you a version of the OS which has the latest Safari, it has 11.1.2. (NOTE: This reinstall over your existing OS does NOT find the online Safari update 13.1.2, would be nice if it did.)
To get the update to Safari, 13.1.2, you need to reinstall a fresh install of High Sierra 10.13.6, presumably only on a disk which does not have an OS installed. Then, after checking the App store, and importantly, waiting for sometime for the server to find the update, you finally get an option to update to Security Update 2020-005, iTunes, 12.8.3, iTunes Device Support, AND finally Safari 13.1.2. This is the only way to get to this update. Strange, because Security Update 2020-005 AND Security Update 2020-006 are both available on Apple's Download page I quoted above. It seems to be an oversight to NOT have included Safari 13.1.2 as a separate download. Or maybe an error, as there are a number of errors on that Download page, missing update numbers, duplicates, wrongly named downloads, etc.
So it may be an error that Apple didn't include it. Or it may be some complication in the update process.
The downside of this is that you need to reinstall every application, unless some clever person can suggest something along the lines of what Apple offers when you upgrade to a new Mac.
Thanks to everyone for their help.