Heterogeneous ice nucleation impacts the hydrological cycle and climate through affecting cloud microphysical state and radiative properties. Despite decades of research, a quantitative description and understanding of heterogeneous ice nucleation remains elusive. Parameterizations are either fully empirical or heavily rely on classical nucleation theory (CNT), which does not consider molecular-level properties of the ice-nucleating particles - which can alter ice nucleation rates by orders of magnitude through impacting pre-critical stages of ice nucleation. The adsorption nucleation theory (ANT) of heterogeneous droplet nucleation has the potential to remedy this fundamental limitation and provide quantitative expressions in particular for heterogeneous freezing in the deposition mode (the existence of which has even been questioned recently). In this paper we use molecular simulations to understand the mechanism of deposition freezing and compare it with pore condensation freezing and adsorption. Based on the results of our case study, we put forward the plausibility of extending the ANT framework to ice nucleation (using black carbon as a case study) based on the following findings: (i) the quasi-liquid layer at the free surface of the adsorbed droplet remains practically intact throughout the entire adsorption and freezing process; therefore, the attachment of further water vapor to the growing ice particles occurs through a disordered phase, similar to liquid water adsorption. (ii) The interaction energies that determine the input parameters of ANT (the parameters of the adsorption isotherm) are not strongly impacted by the phase state of the adsorbed phase. Thus, not only is the extension of ANT to the treatment of ice nucleation possible, but the input parameters are also potentially transferable across phase states of the nucleating phase at least for the case of the graphite/water model system.