electromagnetism - Are the field lines the same as the trajectories of a particle with initial velocity zero

Is it true that the field lines of an electric field are identical to the trajectories of a charged particle with initial velocity zero? If so, how can one prove it?

The claim is from a german physics book from Nolting "Grundkurs theoretische Physik 3 - Elektrodynamik" page 51, let me quote:

Man führt Feldlinien ein und versteht darunter die Bahnen, auf denen sich ein kleiner, positiv geladener, anfangs ruhender Körper aufgrund der Coulomb-Kraft (2.11) bzw. (2.20) fortbewegen würde.

In english:

One introduces field lines and means trajectories along which a small, positively charged, initially resting body moves due to the Coulomb-foce (2.11) resp. (2.20).

2.11 is just the coulomb law, 2.20 is $F = q E$.

(If someone has a better translation, feel free to edit it).

I don't see why this should be true. So it would be great to see a proof or a counterexample with solved equations of motion.

For a magnetic field this claim is obviously wrong since the Lorentz Force depends linearly on the velocity.

Are there other physical fields where the claim is analogously true?

Edit: The answers show that the claim is not true in general but holds in the special case of a highly viscous medium. Is this also the case for moving charged cotton along the field lines in air, as shown in this animation: http://www.leifiphysik.de/web_ph09_g8/grundwissen/01e_feldlinien/01e_feldlinien.htm ?

Do you have any references or more details for this viscous media limit?

Do you have any computational counter example why it doesn't hold in general or a simulation which shows that?