PIR driving TIP120 goes high when the TIP120 is powered

The PIR is essentially a switch with one side connected to Vcc and the other to the output. It is closed on detection of movement; when NOT detecting it is open - so the output is an open circuit and does NOT conform to the requirement for a digital input to the arduino.
That is why you need a pull down resistor between OUT and GND - so when the switch is o/c the digital output is a logic 0

as explained here

int LEDout = 12;  ?????