I think you’re confusing moon phases with lunar eclipses. A gibbous moon appears less than fully lighted because part of the moon’s surface is in the MOON’s shadow, not the earth’s. During a lunar eclipse, the edge of the directly illuminated part of the visible surface of the moon does, as you expect, always appear concave.

Waxing means that it is getting bigger. Gibbous refers to the shape, which is less than the full circle of a Full Moon, but larger than the semicircle shape of the Moon at Third Quarter

At 5.45 on the evening of 17 September I was flat on my back on my Mutare garden lawn taking some hand-held shots of the Waxing Gibbous Moon which was directly overhead when a pair of chanting hornbills flew past. Was lucky to catch one of them with the Moon. Camera: Panasonic Lumix DMC-TZ60 in sunset scene mode.
https://uploads.disquscdn.com/images/826b0c5500ba1ee14e62f6562432434e91e928051be9995a1996ae54c43d8a8b.jpg

Take a ball and a flashlight into a dark room. Hold the ball arms length directly in front of you. Now shine the flashlight on the ball while holding the flashlight above your head and off to the right. Look familiar? The picture above shows moon when the sun is behind you, above and to the right of your view of the moon. If the Earth were casting a shadow on the moon (an eclipse), the curve of the edge of the shadow in the above picture would be opposite; the curve of the edge of the shadow would bend INTO the moon, not AWAY from the center. Hope that helps!