"gibbous"
Not exactly. The lighted side of the moon as seen from Earth is that part that is not shadowed by the Earth.
New moon.
That's the 'Full Moon'.
The moon phase when none of the lighted side can be seen is called a new moon. During this phase, the moon is positioned between the Earth and the Sun, so the side of the moon facing Earth is not illuminated.
half moon
First Quarter and/or Third Quarter.
The phase you are referring to is called "waxing," where the amount of the lighted side of the moon that can be seen increases as the moon moves from new moon to full moon.
That would be Third Quarter.
The waning Moon phase in which only one-half of the lighted side of the Moon can be seen from Earth is called the Third Quarter Moon. It occurs when the Moon has completed three-quarters of its 29.5-day cycle.
The moon phase when none of the lighted portion can be seen from Earth is called a new moon. During this phase, the moon is positioned between the Earth and the Sun, so the side of the moon facing Earth is not illuminated.
Last quarter.
That's "First Quarter", roughly 7.5 days after New Moon.