Are you holding Option at boot, so you can choose the EFI partition to load first, then continue to hold option, so you then can choose the icon for your installer?
Both of my 2012 minis do something similar, so I have learned to continue to monitor an install, so I can always jump on the Option key. Otherwise, the installer will sometimes bypass the EFI boot, and the installer will then boot incorrectly.
One other point - your mini is more than 10 years old. Be sure that part of your "resuscitation" is replacing the battery, which requires pulling the logic board. The battery is on the bottom side of the logic board. (It takes a BR-2032). A dead, or almost dead battery can cause unusual symptoms, and a fresh battery may be part of your solution.
