I’m having difficulty understanding the Quick Start directions for transferring data via direct wired connection f old to new phone: Step 2 says “Plug the Lightning to USB Cable into your new iPhone, then connect the other end to the adapter.” But th

The iPhone 12 mini has a lightning port just the same as old iPhones. It's just the other end that has a USB-C connection. Use the cable from your iPhone XR.

You don't need to connect the two phones by cable at all to use Quick Start. But the old phone needs to be at iOS 12.4 at a minimum.

All this time and I never realized it was possible to do this. I stand corrected and learned a new trick today.

I messed up, but it's nice of you to say thanks! And good luck getting everything migrated over to your new phone.

This support article - Use Quick Start to transfer data to a new iPhone, iPad, or iPod touch - Apple Support - states you need a Lightning to USB 3 Camera Adapter and a Lightning to USB Cable. The Lightning to USB cable should be easy to come by as you should have received one with your last iPhone purchase. The Camera Adapter you'll most likely need to purchase unless you already have one or you know someone that has one already.

Quick start is not wired. It uses Wifi to transfer data from the previous phone to the new phone. There is no facility to connect one phone to another to transfer data with a cable.

Do you own a computer with iTunes?

See this link for the ways to move from one phone to a new phone --> Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support