Action-2D electronic spectroscopy is emerging as a powerful technique to investigate exciton dynamics in molecular aggregates and nanostructures. While maintaining the power of highlighting coherent evolution between the laser pulses, action detection is based on measuring the incoherent signal proportional to the excited-state populations generated by an additional laser pulse. Numerical simulations of the action signal play a crucial role in aiding the interpretation of the spectral features, which may differ from those of the analog coherent technique in a non-trivial way. We present a numerical investigation of the action response of a model of quantum dot as a case study to unravel the exciton and biexciton contributions in the 2D-spectra of nanostructures. The simulation protocol is based on a non-perturbative treatment of the light-matter interaction by solving the Lindblad quantum master equation and the different contributions to the non-linear response are disentangled using a phase-modulation scheme. We analyze how the relative weights of the exciton and biexciton signals determine the lineshape of the spectrum, how they depend upon the physical nature of the detected signal, i.e., fluorescence or photocurrent, and on the relaxation dynamics during the detection-time. Compatibly with the experimental conditions, the choice of the detection-mode and the use of time-gating may eventually facilitate the evaluation of relevant parameters, such as the biexciton binding energy and the timescale of the biexciton relaxation.