Arduino MEGA 2650 R3 pin 13 LED is always on, unless I connect a multi-meter

I have a problem with the on-board LED staying lit up despite running a blink program.

I was running the blink program below, which writes HIGH/LOW to pin 13, but the light was solid. As soon as I connected the circuit between ground and pin 13 with the multi-meter, the LED started blinking.

Here's a video showing my setup. The video shows the "L" LED lit solid. The digital multi-meter (connected to pin 13 and GND) is switched from OFF to DCV, at which point the LED starts to blink. The wire connections are removed, and the LED is again lit up solid.

When the pin is set HIGH, the voltage reading is 5.07V. However, when the pin is set LOW, the voltage reads 0.61V whereas other pins read 100 times less, around 6mV.

void setup() {
  pinMode(13, OUTPUT);
}
void loop() {
  digitalWrite(13, HIGH);
  delay(2000);
  digitalWrite(13, LOW);
  delay(2000);
}

The question is why is this happening? Is my board bugged?

Reference schematic linked here. "arduino-mega2560_R3-sch.pdf"