How do you download High Sierra?

I'm trying to download an installer in order to upgrade my iMac (27-inch, Mid 2010) with OS X El Capitan 10.11.6 (15G22010) to High Sierra.

Problem: High Sierra doesn't appear in the App Store on my iMac with El Capitan. If I search for it, it gives 0 search results.

When I use a MacBook with a newer OS, it tells me that I can't download High Sierra either.

So how can I download High Sierra from the AppStore, even when Apple tells me that I should be able to (giving me a link which is a dead end) - and my iMac is a version which can upgrade to High Sierra?

I understood Steve Jobs' motivational speech that shaving off miliseconds of the booting time of a Mac could be seen as "saving lives". So imagine how many Mac users are trying to update their OS and are wasting hours of their time, for something which should be as easy as it gets (I mean, how common is this?)