What was your last OSX installed on the Mini? If your last OSX was either Catalina, Mojave or High Sierra, then it's more likely that the internal drive was formatted APFS. This would explain why you have 2 partitions. APFS will have 2 and 1 of them can not be unmounted. Also, it's more than likely that your boot firmware on the Mini was updated as well, which means it changes the boot up parameters. The new boot firmware puts 1st boot priority on Thunderbolt and Firewire drives, network boot and the least so with USB key drive, unless it has the right boot firmware that allows secondary boot up from a USB port. Lack of Thunderbolt or Firewire drives or the right USB drive will then force you into the Recovery mode. But you can't install the stock OSX that came with the Mini, which was Mountain Lion (10.8), because it can not read nor write an existing APFS partition. APFS starts from High Sierra and up. Which is what you are seeing now; you're in a rock and a hard place.
So the only way you can install an operating system, preferably Mojave or Catalina is to create an EFI boot USB key drive with the proper Mojave/Catalina EFI boot install so you can select it from the partition manager when you press the "OPTION" key. It should work, but it will take "LONGER" to show up because it cycles through Thunderbolt, Firewire, Network and then USB.
Hope this helps.