There are quite a few papers who, e.g. for the p-Laplace equation, show that weak solutions under some assumptions are not quite in $C^2$ but at least in $C^{1,\alpha}$ (locally). But I am wondering how this is possible as for this the weak solution, let's call it $u$, would have to be classically differentiable with its gradient $\nabla u$ being locally Hölder continuous. But how can a Sobolev-function be classically differentiable? Or does $C^{1,\alpha}$ in context of weak solutions always mean the weak derivative even though the notion $C^{1,\alpha}$ usually means classically differentiable with the derivative being Hölder continuous?
I hope the question is clear. Thanks for help!
Provided I understand your question correctly, I would say that the point of such regularity results is precisely to prove that the weak solution, say $u \in W^{1,p}(\Omega)$ to the problem $- \Delta_p u = f$ enjoys more regularity than required by the mere definition of being a weak solution.
Note that saying that $u \in W^{1,p}(\Omega)$ indeed corresponds to saying that $u$ has a weak derivative say $u' \in L^p(\Omega)$. But this does not prevent $u'$ from actually also belonging to $C^0(\Omega)$, so that $u$ is classically differentiable. Here, the results you mention also prove that $u'$ has Hölder regularity.
Maybe what you are missing is the following type of argument, say in 1D to keep things simple: if $u \in L^2(0,1)$ has a weak derivative belonging to $C^0(0,1)$, then $u \in C^1(0,1)$.