Understanding the "between" lunar phase names is easy if you break out and define 4 words: crescent, gibbous, waxing, and waning. The word crescent refers to the phases where the moon is less than half illuminated. The word gibbous refers to phases where the moon is more than half illuminated. Waxing essentially means "growing" or expanding in illumination, and waning means "shrinking" or decreasing in illumination.
The Waxing Gibbous moon is the phase when the moon is becoming visible: This phase occurs the first quarter, the sunlit portion is still increasing, but now it is more than half, so it is waxing gibbous.