Confusion regarding work and the first law of thermodynamics
The general formulation of the first principle for a closed system says that $$L+Q= \Delta K + \Delta U + \Delta u$$ Where $L$ is the total non-conservative work done on the system. $Q$ is the heat entering the system. $K$ is the macroscopic kinetic energy, $U$ the macroscopic potential energy and $u$ the internal thermodynamic energy. Usually $U$ and $K$ are disregarded, the former because $-\Delta U$ can be viewed as further work on the system due to macroscopic conservative forces, the latter in particular because one usually deals with initial and final states where all macroscopic parts of the system are at rest. Exploiting this equation the processes you consider can be coherently discussed. In particular, if your action on the stone does not imply deformations of its form with production of internal dissipative stresses and there is no flux or production of heat (a completely mechanical kick), there is no variation of $u$ but only of $K$.