Just go to the Moon and take a look. From the Moon's point of view, the Earth is a selenostationary$^{1}$ satellite just as you describe. And as others have said, it goes through the full range of phases every lunar day.
$^{1}$ "Seleno" refers to the moon so, just as geostationary means orbiting Earth above the same point, selenostationary means orbiting at a particular point above the moon.