LC oscillator versus Kirchhoff's law for meshes
I know that the vector diagram is not exact, because what I consider
  here, is a transient process with decreasing amplitudes while the
  vector diagram represents stationary behavior with constant
  amplitudes.


Not just inexact but wrong.


  However, that shouldn't make a difference, because the phase angles of
  the voltages are correctly represented, aren't they?


You CANNOT use a phasor diagram for this - phasor diagrams are reserved for steady state AC analyses (with a constant excitation frequency). For instance, if you apply a DC level of 12 volts across an inductor, current rises at a rate of V/L - where is the 90 degrees phase shift here? No, please don't use phasor diagrams for transient analyses.