Need to install High Sierra on older iMac: mod 1311. Working from current MacMiniPro

OnTheRez Said:

"Need to install High Sierra on older iMac: mod 1311. Working from current MacMiniPro: Install High Sierra on different device: iMac mod. A1311. Clean format. WiFi on. Attempt to download HS. Message: ‘Can’t reach server.’  Known Apple error. App Store won’t allow HS download to this MacMiniPro ‘Not compatible with this computer.’ Can’t download to external drive. I'm stuck.[...]"

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Getting a Connection to Apple to go Through...

A. Reset the SMC and your NVRAM:

Sometimes when changes are made to the system(i.e. updates), system configurations (i.e. for the keyboard) become manipulated, technically. So, reset the SMC and NVRAM.

B. Use Ethernet:

Connect your Mac using Ethernet.

C. Create and Use a USB Bootable Installer:

If this is a clean install, see about creating an using a USB Bootable Installer: Create a Bootable Installer for macOS - Apple Support

D. Use Internet Recovery Mode:

Connect using Internet Recovery Mode, downloading it through WiFi or Ethernet.

A1311 incorporates several different model number iMacs. Models that shipped with OS X installation media on optical discs require those two discs from cradle to grave. If you misplaced them / can't find them / etc things get difficult in a hurry.

iMacs that shipped beginning on or around August 2011 are also model number A1311, and shipped with no discs. Apple silently and inexplicably removed the ability to contact the High Sierra download through their Recovery servers a year or so ago, marking the first time in Apple history they effectively abandoned support for customers who wish to restore their Macs to their original condition (a requirement for sale, among others) leaving them with no ability to install an operating system: as you wrote, stuck.

It will not be productive to discuss or determine which specific model you have; let's just assume you don't have those discs. The workaround for the known inability to contact Apple's Recovery servers for High Sierra and earlier OS is discussed here among other references that may be found on this site. Be advised I have not tried it, but the procedure seems reasonable to me.