fundamental astronomy - What exactly is 'eastward movement'

From the point of view of an observer hovering above the Earth's surface, the rotation of the Earth is eastward. That is, points on the surface appear to move in an easterly direction as the Earth spins.

Extending this to the orbital motion, there are at least two possible interpretations. One is that the spin angular momentum and orbital angular momentum are in the same direction. This is roughly true (and certainly closer to true than the opposite). Another interpretation would be that the Earth moves around in its orbit such that it heads in an easterly direction when referred to a longitude on the Sun. This is also approximately true.

If you are asking what is a general definition of the north pole, then I suppose it could be the direction of the angular momentum vector. Then yes indeed, the right hand grip rule tells you the spin is eastward as it relates to north.