In the below picture, what exactly is the streamline? Is it the three blue curves or is it the E vector drawn? Also, to calculate the equation of the streamline, we are using the ratio of the components of the E vector and setting that equal to the derivative at that point---is that correct?
To answer your question, the "streamlines" are the blue curves, and that does appear to be an accurate interpretation of the caption.
That being said ... is this, by any chance, an introductory calculus textbook? I have never seen an integral curve of the electric field referred to as a streamline. They are usually called electric field lines, while the word streamline typically refers to an integral curve of the velocity field of some fluid. It's not strictly wrong, as the two objects are mathematically identical constructions, it just doesn't sound like something you'd find in a physics book.
Also, that is not typically how one solves for them because it runs into issues when $E_x=0$. The idea is more or less correct, but it's implemented differently to avoid this problem.
