Is every electric wave is an 'electromagnetic wave'?
Why we only assume the electric field only?
In free space their is a fixed relationship between the electric and magnetic fields, determined directly from Maxwell's equations: the two fields are oscillate at the same frequency, maintain their phase relationship, and are orthogonal (perpendicular) to each other in their oscillations.
So for many purposes one only needs to look at the electric field, which is easier to measure.
In a wave guide the situation is different, and depending upon the boundary conditions the modes can be TE, TM, or TEM. These are not studied in elementary physics or optics, but as they are important parts of our technology, they are covered in more advanced courses, and in the relevant areas of engineering.
But the direct answer to your question is no; if the electric field is changing, which is required for it to be a wave, it will itself generate a magnetic field, so they are always found together. But if one or the other is phase lagged, or more strongly absorbed, interesting things occur.