...and my understanding is....light is measured to travel at the same speed regardless of the frame of reference. It would not matter whether the star is moving towards or away from you, you would measure light to travel the same speed. It does not matter where or how the speed of light is measured; it will always be the same.
When light passes through some non-vacuum medium, it isn't slowing down, it's travelling farther, because the gravity of the matter bends space-time. The light has to travel farther because the path is no longer straight.