first of all, the quantum mechanical description 1 in its current form is impossible. A charged particle can't simply absorb a real photon. This is most easily seen in the final particle's rest frame. There is no photon left so the total energy is just the rest mass of the particle, times $c^2$, but the initial state has a higher energy of the particle in this frame, because the particle was moving, and also an additional positive energy of the photon. So the energy couldn't have been conserved in this process.
A charged particle may only absorb a real photon if it emits another one in another direction. So microscopically, it's always a process composed out of the Compton scattering subprocesses. If the number of real photons in the same state is large, they may be described as a classical wave. You may partially quantize the system, keeping the electromagnetic field classical and quantizing just the particle, or otherwise. The reason why this description agrees with the full quantum description in the classical limit is manifest.
Also, I don't quite follow the difference between your classical description and the quantum mechanical description 2. Concerning the usage of virtual photons, well, if you want to study the whole process using the tools of quantum field theory - including the pre-history when no electromagnetic wave existed - then of course the electromagnetic wave had to be created at some point, and some of the photons were absorbed by the charged particle. The charged particle became virtual for a little moment as well, before it emitted another photon which is needed for the energy conservation, as I have explained.
So the photon that was absorbed by the charged particle was virtual - it only existed for a finite amount of time. However, the electromagnetic wave was probably propagating for such a long time that even this photon may be called "real". There is a simple relationship between virtual and real particles - real particles are the virtual ones that happen to sit exactly on the mass shell, so they satisfy $E^2-p^2=m^2$. This identity may be exactly checked only when $E,p$ are measured totally accurately - which means that the particles must exist indefinitely. If they don't exist indefinitely, then they're always "virtual" to some extent, but chances are that if they exist for a long time, you may also imagine that they're "real".
The "virtuality" of a particle may be defined as the difference $E^2-p^2-m^2$ - the distance from the physical mass shell. If the virtuality is low, the virtual particle may exist for a long time and look "real".
Finally, there are no "virtual photon states" in the Hilbert space. The Hilbert space only contains real particles. Virtual particles are an object that appears in the calculation of probability amplitudes for various processes - in the Feynman diagrams. Virtual particles are internal lines of Feynman diagrams, given by propagators that determine the 2-point function (correlator) of a quantum field. But they don't correspond to any physical states. There are no off-shell physical states in the Hilbert space.
So if you have a history in which some particles exist for a finite amount of time, so that they're strictly speaking virtual from the Feynman-diagrammatic viewpoint, it is still true that at every moment, there must exist some real particles that are actually present. However, it is tough, misleading, ambiguous, and unnecessary to calculate the "exact intermediate states" in quantum field theory. Such objects - wave functionals - would also depend on the field redefinitions (of the quantum fields), renormalization schemes, and other things. It's actually very useful to avoid these things when they're not necessary and only talk about the things that can be measured - the cross sections that may be calculated from the scattering amplitudes.
A problem with the "wave functionals" of the intermediate states is that they're only well-defined with respect to a reference frame - but virtually all regularizations we know to calculate the loop diagrams rely on the Lorentz symmetry. Because the Lorentz symmetry is obscure by the foliations of the spacetime, it becomes harder to "regulate" the exact wave functional at the loop level. Of course, at the classical or semiclassical level, one may describe very accurately what's going on.
In your particular situation, there was no real problem because all the photons in the problem were really on-shell, and you may present them as real photons if you wish.
Best wishes
Lubos
There is overlap with other questions linked in the comments. But, perhaps the focus of this question is different enough to merit a separate answer. There are at least two distinct but equivalent formalisms of QFT, the canonical approach and the path integral approach. Although, they are equivalent mathematically and in their experimental predictions, they do provide very different ways of thinking about QFT phenomena. The one most suited for your question is the path integral approach.
In the path integral approach, to describe an experiment we start with the field in one configuration and then we work out the amplitude for the field to evolve to another definite configuration that represents a possible measurement in the experiment. So in the two slit case we can start with a plane wave in front of the two slits representing the experiment starting with an electron of a particular momentum. Then our final configuration will be a delta function at the screen representing the electron measured at that point at some later specified time. We can work out the probability for this to occur by evaluating the amplitude for the field to evolve between the initial and final configuration in all possible ways. We then sum these amplitudes and take the norm in the usual QM way.
So in this approach there are no particles, just excitations in the field.
Best Answer
Weinberg is right.
The issue here is with the usual interpretation of the wavefunction as an amplitude density. This implies being able to localize the particle in an arbitrarily small region. However, it is not possible to localize photons (or any massless particles with spin, for that matter).
The reason for this is the careful definition of what localization means mathematically. It means that there must exist a projection operator with certain properties that intuitively correspond to the idea of measuring a particle at a given location. For a massive particle, one can show such a projection operator exists by looking at the little group (the subgroup of Lorentz transformations that leave the "rest frame" invariant). Because there is a rest frame for the massive particle, the little group is $SO(3)$, the group of three dimensional spatial rotations. If you "quotient out" the little group you're left only with boosts, and since the space of boosts is homeomorphic to $\mathbb{R}^3$, you can use them to define a position operator. A particle being "localized" then just means that you're not allowed to perform translations without changing the description of the physical state. In other words, localizing a particle breaks translational symmetry. So far, so good.
For a massless particle, there is no rest frame, so you must say the particle's momentum lies along a spatial direction and consider what transformations leave the momentum invariant. Then, the little group is $ISO(2)$, the group of translations and rotations in the plane orthogonal to the momentum. Already we start to see the problem: the little group is "intruding" on the possible characterizations of position states. This is no problem for a spinless particle -- intrude away -- but for a vector particle the translations correspond to gauge transformations, which means you can't project out states that break translational symmetry without also breaking gauge invariance -- a big no-no. So a photon can't be conventionally localized, with a Maxwell field for a "wavefunction" or anything else.
A more heuristic way of saying the same thing is to imagine multiplying the Maxwell field by a position operator. In the vacuum it's supposed to be divergenceless, but any scalar function that depends on the position breaks this condition. What Wightman has shown is that it's impossible to construct a position operator consistently, scalar or otherwise.
I have bastardized a long mathematical story so I encourage you to read the original references. If you're not familiar with the little group and classifications of particles under the Poincaré group I recommend you start with chapter 2 of Weinberg Vol. 1. Then read here for Newton and Wigner's original proof, here for the more general construction by Wightman, and this and this for weaker notions of photon localization that actually make sense but rule out interpreting the Maxwell field as a wavefunction.
PS: In case you're wondering, Weinberg's second remark that the Klein-Gordon field cannot be interpreted as a wavefunction is also correct. Here however we're led to the standard story about negative energy states and propagation outside the light cone that you can see in pretty much any QFT textbook.