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Good question. What the article means, though, is that what we normally notice each night and year, where stars rise and set, some stars move forward and backward relative to background celestial objects, and the constellation positions change over the months, is because the Earth is rotating on its axis and revolving around the Sun, changing our view.
However, you are right in that stars do move. For example, binary stars orbit around a common center of mass. Some stars slowly orbit around the center of their rotating galaxy; an example of this would be our own Sun, revolving around the center of the Milky Way galaxy (completing one orbit in about 225 million years).

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