feynman-diagramsparticle-physicsquantum-electrodynamicsquantum-field-theoryscattering
In the tree level, the $e^-e^- \to e^-e^-$ scattering has two Feynman diagrams, the first one is indicative that one electron emitted a photon which was later absorbed by the other electron:
However, I have yet to understand how we can interpret the other term:
at first glance it would seem like the electrons changed places or simply that they have opposite 4-momentum compared to the previous diagram, but I am still not sure what is the correct interpretations.
When drawing Feynman diagrams, it is important to fix the incoming and outgoing particles and their momenta. For the inexperienced, this is ideally done before drawing the rest of the diagram in order to avoid confusions like yours. So, let's assign each state some momentum: Let's give the electron in the upper left corner (4-)momentum $p_1$, lower left $p_2$, upper right $p_3$ and lower right $p_4$. Let us consider all momenta as 'flowing' from left to right (this is an arbitrary choice that makes no difference to the actual calculations).
Then, it immediately becomes clear why the two diagrams are different. For instance, check out the momentum on the photon line (call it $q$, and imagine it flowing top to bottom) that connects both vertices. Conservation of (4-)momentum forces always applies, so we can write it in terms of the momenta flowing into the vertices. For the $t$-channel diagram, it is clear that $q=p_1-p_3=p_2-p_4\equiv \sqrt{t}$. However, for the $u$-channel diagram $q=p_1-p_4=p_2-p_3\equiv \sqrt u$. Thus, the photon carries a different momentum. When carrying through the calculation, it will be seen that this makes a serious difference.
There is a more general point to be made here: Feynman diagrams can be very misleading, especially to those that are not (all too) familiar with the actual calculations that they visually represent. When it really comes down to it, it is of vital importance to realize that the diagrams should always be considered as secondary aids in performing calculations, which can just as well be done without drawing anything (it just takes a lot longer): Always be very careful when interpreting the diagrams.
I'm assuming you're using the lagrangian
$$\mathcal{L}=\overline{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{\psi})\psi+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}-g\overline{\psi}\psi\phi,$$
or something similar.
The diagrams don't require you to have a particle-antiparticle-scalar triple to make a vertex. It just requires that all interaction vertices must have two $\psi$ propagators and one $\phi$ propagator. So long as the charge flow into each vertex is valid, you are fine. For instance, think of the first process,
$$\psi\psi\to\psi\psi$$
as a $t$ channel process. It's just the
$$\psi\overline{\psi}\to\psi\overline{\psi}$$
diagram, but rotated on its side (that is, related by a crossing symmetry).
The $\phi\phi\to\phi\phi$ process is actually a bit more complicated, and there isn't a diagram to second order that contributes to it. The lowest order diagram would be a box diagram (see below, with curvy lines representing $\phi$ propagators and straight lines representig $\psi$ propagators).
I hope this helped!
Best Answer
It is precisely what you said. When you do a scattering experiment, you're throwing in two electrons with momenta $p_1$ and $p_2$ and see two electrons coming out with momenta $q_1$ and $q_2$. However, electrons are indistinguishable, so you can't know whether the electron with momentum $p_1$ is the one with momentum $q_1$ or the one with momentum $q_2$ (to be fair, due to indistinguishability, the question doesn't even make that much sense). The first diagram can be thought of as pictorially describing the case in which $p_1$ becomes $q_1$ and $p_2$ becomes $q_2$, while the second diagram describes the possibility of $p_1$ becoming $q_2$ and $p_2$ becoming $q_1$.
It should be remarked that Feynman diagrams should not be taken too literally. They are mainly just computational tools and interpreting as what actually, physically happens is an extra philosophical step. The surely provide a pictorial interpretation, but it is important to recall that there is nothing ensuring that is what actually happens. Some physicists do prefer to interpret it like that, it is important to notice this is an extra philosophical step (just like choosing to interpret as something with no physical meaning).
Also, while splitting the two diagrams is necessary in the computation and this allows the interpretations I explained in the first paragraph, it is important to recall electrons are indistinguishable, so there's no really way of saying (or even asking) where the electron $p_1$ turned into $q_1$ or not.