The book is imprecise because there are other types of polarized light. Now, consider a point in the path of the wave, like the green point in animation you mentioned. In that image, the light is linearly polarized so the electric field draws a line at the location of the point. However, there is also circularly polarized light where the electric field draws a circle as in this picture
Then, light is said to be polarized if the electric field at a given point draws a line, a circle, or in general, an ellipse. Unpolarized light occurs when you have many of these waves superimposed which results in an electric field that oscillates randomly in all directions.