OS X Installer files are automatically deleted from Applications as part of the install process. Normally, it shouldn't need moving there, App Store ought to put it there directly as part of the download process.
They are very unlikely to have any 'related files' of the type that AppCleaner could find.
If you did want to keep a copy for the future, then you need to copy it outside Applications before running the Installer.
Massive bump to @IlyaB, I was stumped trying to get the full 5gb Install MacOS High Sierra.app but was stuck with the 19mb stub version.
My Process to get a working app that is currently making my bootable USB for High Sierra
1: Go to Mac App Store and download High Sierra (the 19mb stub)
2: Click Continue in the installer app once it downloads
3: Choose your boot disk (or any disk that has space)
4: Allow the installer to download (this is the contents of the missing "SharedSupport" folder within the installer stub)
5: The installer will prompt you to restart after finishing the 5gb download, at this point quit the installer and do not restart your system!
6: Go to your root (or whichever location you chose in step 3) and find the folder "macOS Install Data"
7: Copy the contents of that folder disregarding .DS_Store
(9 files, folders, dmgs, pkgs, and plists)
8: Go to your Applications folder where the Mac App Store downloaded the stub version of "Install macOS High Sierra"
9: Right Click on that file and "Show Package Contents"
10: Within the Contents folder, create a new folder called "SharedSupport"
11: Paste the files you copied earlier from the installer download
12: And you're done! You should have a fully functional Installer file for bootable USB drives or just copying to other systems whether they be offline, or you just don't want to download the installer over and over.
I am currently in the lengthy process of creating a bootable USB drive to make a Coffee Lake i7 8700k Hackintosh build. I'm following this guide with the recommended motherboard. I'll try to update here with my success/ failure.
Best Answer
Default Location
When the macOS High Sierra Installer is downloaded from the Mac AppStore, it, like any other application downloaded through the Mac AppStore, ends up in the Applications folder. The name should be "Install macOS High Sierra.app" (or its equivalent in your language).
Typical Auto-Installation Prevention
Since the first Mac AppStore-based Operating System (Mac OS X Lion) was available to download, the way to prevent an automated installation upon completing the download was to cancel (quit) the installer, open the Applications folder and move (not copy) the installer to another location such as the desktop. Having done this every time, including High Sierra (in September 2017), I can verify that it worked then too.
Why might it be beneficial?
Apple has always been known for staying top of security, rather than playing a slow game of catch-up. It's thus only natural that the latest macOS version is somewhat "forced" onto the user's computer. Afterall, for the majority of users, making the conscious decision of clicking the download button means they intend to install it right away. Unfortunately, though, there are many people who think that clicking the download button is all there is to it - that's where the application opens and a window pops up to tell them otherwise, showing them what must be done. To you and me, this is very annoying because we don't fall into the "negligent" category, but just imagine if people think they have the latest version and are up-to-date with Apple's latest technologies and security only to find out they were just half way there.
What could you try next?
I think we can both agree that if it hints at the High Sierra option at startup, it's most likely that High Sierra has been downloaded and does indeed reside somewhere on the mac.
ADDITIONS
Please upload an image of the prompt you get upon restarting.
Perhaps something went wrong during the download procedure and the installer is not complete as a whole, yet some part of it thinks it is (perhaps a property file was written to too early), making it seem as though the update is ready to install when in fact isn't.
For this, I would first like you to navigate to the App Store folder at
/Users/<USERNAME>/Library/Application Support/AppStore
and check that directory (including subfolders) for anything that may be High Sierra related. You may wish to consider this article to resolve download issues. It may seem a little far fetched but worth a go if all else fails.Check under the Purchases tab in the Mac App Store, that High Sierra actually appears. Are you sure you downloaded it, or are you simply getting notifications that the upgrade is available (as your comment may hint at)?
Let me know how you make out and I'd be more than happy to assist further.