The lines of force represent the direction a free positive charge would move if one was present. The reason the lines of force are in the outward direction from a proton is because a proton will repel a proton, and thus move outwards. The reason the lines of force are inwards for an electron, is because an electron would attract a proton, thus the proton would move inwards.
So to summarize, force lines are defined to be in the direction a POSITIVE charge would move, if one were present. This is just a human convention/definition.
An electron has the opposite charge properties to a positive charge, and a free electron will move in the direction that opposes the force lines. So an electron will move away from another negative charge, and towards a positive charge.
ELECTRON-PROTON ATTRACTION: a simple, semi-classical analysis to avoid full scale QFT.
The exchange of the photon between the proton and the electron leads to attraction, only because the total energy of the electron is negative.
Let us consider the hydrogen atom for simplicity, and imagine the funnel-like shape of the electron energy. The total energy of the electron in the hydrogen atom at distance $r_0$ from the proton is
$E(r_0)=-\frac{e^2}{8\pi\epsilon_0 r_0}$.
If the distance $r_0$ is sufficiently short, then the electron will emit a photon which will be absorbed by the proton, and the amount of energy of the exchanged photon will be dictated by the uncertainty principle:
$\Delta E\Delta t=\hbar.$
But $\Delta t=\frac{r_0}{c}$ so that
$\Delta E r_0=\hbar c\rightarrow \Delta E= \frac{\hbar c}{r_0}$
So the new energy of the electron will be
$E_1=-\frac{e^2}{8\pi\epsilon_0 r_0}-\frac{\hbar c}{r_0}=-\frac{e^2+8\pi\epsilon_0\hbar c}{8\pi\epsilon_0r_0}$
or the equivalent amount of energy corresponding to some new position $r_1$
$-\frac{e^2}{8\pi\epsilon_0 r_1}=-\frac{e^2+8\pi\epsilon_0\hbar c}{8\pi\epsilon_0 r_0}$
from which we get $r_1$ in terms of $r_0$
$r_1=r_0\frac{e^2}{e^2+8\pi\epsilon_0\hbar c}<r_0$
Therefore the electron moves closer to the proton rather than farther from it (an attractive force.)
Best Answer
You are correct; your teacher is wrong.
Consider protons and electrons moving parallel in the same direction and with the same speed. In the inertial frame of the charges, we clearly have an attractive electrostatic force that will make the beams bend towards each other, and no magnetic force.
The attraction will be there also in our frame of reference, in which we will measure both a slightly higher electrostatic attraction and a small magnetic repulsion (which in the end will give the same behavior).