The key under the stick

A switch and a pull-down

The stick is also a button. Push it straight down, without tilting, and it clicks: a small switch under the housing closes and joins KEY to 3V3. Let it up and the switch opens.

An open switch leaves KEY connected to nothing, and a pin connected to nothing reads whatever the air around it suggests. So the board adds R1, a 10 kΩ resistor from KEY to GND at the bottom left, by the stick's legs. With the switch open R1 holds KEY at 0 V; with it closed the switch wins, because it is a direct wire to 3V3 and R1 is 10 kΩ.

A resistor that holds a line at 0 V until something lifts it is a pull-down. Most buttons are wired the other way, with a pull-up and a switch to GND, so they read HIGH at rest and LOW when pressed. KEY is LOW at rest and HIGH while clicked. Test for HIGH.

INPUT, not INPUT_PULLUP

Because the board already has its resistor, the pin needs none of its own:

pinMode(KEY_PIN, INPUT);        // R1 on the board pulls it down

INPUT_PULLUP switches on a resistor of tens of kilohms inside the chip, from the pin up to the chip's supply. Against R1 it makes a divider, and at rest KEY no longer sits at 0 V but some way up towards the line where LOW becomes HIGH. It will usually still read LOW, but with less margin than it should have, and on an Uno the pull-up goes to 5 V, so every click also feeds a little current from 5 V into the board's 3V3. Neither is worth it.

3.3 V on an Uno is still HIGH

On an Uno, KEY's HIGH is 3.3 V from the header, not 5 V. The Uno's chip, an ATmega328P, counts anything above 0.6 × its 5 V supply, 3 V, as HIGH, so 3.3 V clears it. It is closer to the line than a 5 V signal would be, which is one more reason to leave the pull-up off.

It bounces

The switch is a pair of metal contacts, and metal contacts bounce: for a moment after they meet they open and close several times. A loop that reads KEY thousands of times a second sees each bounce as a click. Count a click when KEY goes from LOW to HIGH, and look again only after a pause; the push button's book shows the pattern.