There is an ambiguity in what is meant by the path of the light.
If you mean the path of any particular photon (i.e., the light emitted at a given time), then your picture is right: This path is straight in the elevator frame by the Einstein Equivalence Principle. However, it is horizontal only if it was emitted at the moment the elevator was at rest relative to the laser. Otherwise, it is tilted because the velocity of the photon has a vertical component in the elevator frame.
If you mean the instantaneous locus of all photons in flight (i.e., the light existing at a given time in the elevator frame), then this path bends downward in the elevator frame -- opposite to the picture you attribute to your teacher. This may be counterintuitive, but consider the elevator after it has been falling at acceleration $-g$ for a time $T$ since being at rest relative the laser. Assume the vertical velocities involved are small compared to $c$, so the horizontal velocity of light is essentially $c$ (up to higher-order terms). 562
In the elevator frame, the photon at horizontal coordinate $x$ (the laser is at $x = 0$) was emitted at time $\tau = T - x/c$, when the laser was at vertical position $+g\tau^2/2$ and had vertical velocity $+g\tau$. Thus the photon, after traveling in a straight line for time $x/c$, has current vertical position $g\tau^2/2 + g\tau(x/c) = g(T^2 - x^2/c^2)/2$. This is a downward-bending parabola. The same effect would be seen in Newtonian physics with, say, a hose emitting water or a machine gun emitting bullets.
Note that the above "paths" are defined in terms of the reference frame of the elevator (notional grid of clocks and rulers), and for the case of light, neither of these paths would be straightforward to observe by what someone naively (optically) sees in the elevator. For example, if there is smoke that scatters the light so it can be seen from elsewhere in the elevator, the instantaneous appearance of the light beam would not correspond to the second interpretation (locus of all photons in flight), because of the differing time for propagation to the observer's eye from different parts of the beam. 562