Mojave can't be installed but still boots to Installer partition I downloaded macOS Mojave from the App Store and, upon running it, it told me that my MBP (15-inch, Mid 2010, currently High Sierra) can't be upgraded to Mojave. 

Then it restarted my Mac... into the same installer again. This time, the installer complained of some missing files.

I booted back into my Mac and deleted the .app file, but the boot menu (by pressing Option during boot) are still showing 3 partitions: mac OS Installer, OSX, and Recovery HD—in that order.

The weird thing is, both mac OS Installer and Recovery HD seem to open the same thing—a menu(?) where you can reinstall Lion(!!!).

My question: How do I remove mac OS Installer from the boot menu? Or, at least, reorder the menu so that OSX comes first.

I read on the internet to rerun the installer and cancel it there—well, I've deleted the file so I have to download it again, but now App Store prevents me from downloading it ("This version of macOS 10.14.5 cannot be installed on this computer."—I know! Why didn't you say so last time!)
You would have to get another Mac with an ethernet port, boot your Mac in Target Disk mode (Command T) and connect it via ethernet to the other Mac. You can also use the Thunderbolt 2 ports to network the two Macs. 

Once it's up, you can access your Mac's HD as if it was an external HD, and you can remove the file from there.

Additionally, you could try booting into recovery Mode (Command R) and fix your HD's permissions with Disk Utility.

The thing is, the App store would have prevented you from even downloading Mojave, since it checks your hardware for compatibility. Mojave also lists its system requirements as well a the compatible systems on the website, as well as the App store.

So, I suppose you must have downloaded Mojave from somewhere else. The reason why we can't install Mojave in older systems is because of compatibility. Mojave also needs a lot of resources that only newer systems can handle. The oldest systems that can still handle Mojave are those released in 2012, whereas those from 2011 and below can only handle up to High Sierra.