"Burn in"? I'd hate that term, too.
"Burnish" is a fine word for that process.
bur·nished, bur·nish·ing, bur·nish·es
1. To make smooth or glossy by rubbing; polish.
2. To rub with a tool that serves especially to smooth or polish.
3. To improve or make more impressive: achievements that burnished her reputation.
Burnishing brakes involves using the shoes or pads to "polish" the drums/rotors, the heat created helps to remove manufacturing solvents and to cure binders in the friction material of the shoes/pads. It establishes an "improved" working surface on both the friction material and the mating iron, steel, carbon fiber or whatever the friction surface of the drum/rotor is made of, especially if the drums/rotors have been resurfaced as part of the "brake job".
For the record, every drum-to-shoe gauge (Lower item in the tool photo) I worked with was a piece of crap. The "points" that set the drum size tended to be worn if not actually bent, and the sizing adjustment was so sloppy as to make precision impossible. I quit using them decades ago. I just go by "feel" when it comes to shoe-to-drum drag. I set the initial "drag" lightly, then step on the brake pedal to align the shoes to the drum as they would be in actual use. Then tighten the adjuster some more. My initial drag may even be heavier than some would permit, because I know that after burnishing, the friction material high-spots are going to be worn away--and it's the high-spots that create most of the initial drag.