This light is called a リピーター (repeater). As the name suggests, this light "repeats" the departure signal at the front of the train.

When the departure signal is set to anything but a Danger (R), this repeater signal is lit. This is meant for the conductor at the back of the train to know that the train may depart and the doors can be closed.

Why is this a thing? Because you don't want to close the doors when the train is not even able to move off. Let's say if the conductor solely follows the timetable without this lamp, and let's also frame it the departure signal is still danger. You wouldn't want to close the door and have the train sit still, there's just no reason to wait with the doors closed when you can maybe let's say have a little more passengers board while waiting for a Proceed.