How to change your primary device for receiving the verification code?
After doing some research online, I was unable to find any mention of a "Primary Device". Any device that is registered to receive Two Step verification codes will receive the code.

I think what you need to do to receive the codes on you Mac-Mini is register the device to receive Two Step Verification codes. 

Step Up device to receive Two Step Verification Codes:


Go to Apple () menu > System Preferences > iCloud > Account Details
Click Security.
Click Turn On Two-Factor Authentication.


Instructions from Apple KB - Two-factor authentication for Apple ID

Hope this helps