Your confusion arise from the definition of East and West hemispheres. That's an arbitrary and confusing definition. Because East and West are relative directions. Meaning they depend on the position were they are specified.
East and West come from the proto-germanic languages, where East means dawn and West means evening. Therefore, East from any given point is roughly the direction at which the sunrise happen. And West is where the sunset happens.
That's why the east/west hemispheres definition is confusing, because if you are in the middle of the pacific, the Western hemisphere is towards the East... confusing.
So let's forget about East/West hemispheres, and consider the East/West cardinal directions. Where East is 90° to the right of North and West 90° to the left of North. These directions keep the original spirit, on which East points roughly to the sunrise, and West to the sunset.
Let's also forget about clockwise and counterclockwise, because that depends on where are you looking from. If you are on top of the North pole it will look like Earth is rotating counterclockwise, but from the top of the South pole it will be clockwise.
Now, when we say that Earth rotates to the East, we mean that each point on the surface is moving to the East of that point.
The following figure taken from pulauubinstories.com will help clarifying the point:
In the figure is clear that the fact that sunrise happens to the East is because we are moving in that direction. At dawn, as we move Eastwards we enter in the sunlit side of Earth, producing the sunrise. And as we keep moving Eastwards during the day, we enter in the night side of Earth, leaving the sunshine and the sunset to the West.
Therefore, as East means "dawn" it follows (basically by definition) that Earth rotates to the East.