The low voltage could have definitely caused the relay to burn out. With lower voltage, the amount of current sourced by the wiper motor would increase in order to draw the same amount of power. Increased current draw by the motor probably killed the relay. In other words, the motor is going to draw X watts. Power (watts) = V*I where V is voltage and I is current. If V goes down, I will go up to satisfy the power requirement needed by the motor.
I'm a retired elevator mechanic and shorted relay coils were not uncommon but I don't remember running into that on a car. Good find Ramon. Always nice when a cheap part fails vs. an expensive one. 🙂 👍👍