A point not mentioned by the article: the electrons in a metal at room temperature are already moving very quickly due to their thermal energy (at the order of 100km/s) — much faster than the speeds quoted in the article, which is what's called the "drift velocity".
This thermal motion is essentially random, and the electrons constantly scatter off the nuclei every which way, so it cancels out and doesn't create a net current.
So, it's less than the electrons gently move under the influence of an electric field, and more that it introduced a slight bias in the existing thermal motion.
E: To clarify in case it may have been unclear, this is unrelated to the speed of propagation of the electric field, which as the article says is the speed of light.
This thermal motion is essentially random, and the electrons constantly scatter off the nuclei every which way, so it cancels out and doesn't create a net current.
So, it's less than the electrons gently move under the influence of an electric field, and more that it introduced a slight bias in the existing thermal motion.
E: To clarify in case it may have been unclear, this is unrelated to the speed of propagation of the electric field, which as the article says is the speed of light.