No joy. My old link happened to be for Sierra, not High Sierra. I will keep looking.
But in any case, it looks (to me) like Apple recently
• screwed up the link in Paragraph 4 of https://support.apple.com/en-us/HT208969 directing it to iTunes instead of AppStore.
• didn't change that page's "Published Date: October 07, 2019" to reflect that change.
• again hid the ability of a Sierra user to upgrade to High Sierra using the App Store.
I located an installer that I had previously downloaded for High Sierra (v. 13.6.02), but it no longer works, resulting in a dialog box saying "This copy of the Install macOS High Sierra 13.6.02.app application is damaged, and can‘t be used to install macOS." -- after it apparently tried to link to somewhere. I am pretty sure it used to work, and I had saved it just in case I needed a quick local reinstall when I did a fresh High Sierra installation on an external drive through my MacBook Air just last week.
Alternative solution? I searched at support.apple.com/downloads for "high sierra". It does not find any real High Sierra installer, only updaters and security updates. The text in the updater links refer the user to the AppStore to obtain the base High Sierra upgrade. Of course, that's where we all started before. Fail.
Frustration continues to reign. I left my dissatisfied comments for Apple on the subject page. I wonder if they will read them.