Why is it that what looks to us like a Half-Moon is called a Quarter-Moon by Astronomers?

Much to our surprise, when astronomers throw lunar fractions around, they are referring to the orbiting cycle of the Moon, not the appearance to us.
The Moon is half lit when it is a quarter of the way around its orbit. The count begins when the moon is in the vicinity of the sun (at "new moon" phase). "First quarter" is when the Moon has traveled one-quarter of the way around the sky from there. The Moon is full when it is halfway around the sky, and the "third quarter" or "last quarter" is when it's three-quarters of the way around its orbit. Laymen sometimes misname other lunar apparitions. Sometimes a crescent moon hanging low in the evening sky is called a "New Moon," but the moon is far from new.
the crescent Moon is some three or four days past the actual moment of New Moon, which is the instant when the center of the Moon passes between the Earth and Sun.
http://www.space.com/moon