[tnecniv]

Your personal one is the correct one from my perspective (as someone that knows a lot of math and is familiar with statistical mechanics as math and not physics). The ODEs that we get from classical mechanics are typically reversible: we can write down an ODE that does the same thing but backwards.

You cannot do that for the PDEs that arise in statistical mechanics and the result is the second law. These PDEs arise from approximating many copies of deterministic systems as continuous distributions of states. Entropy is not a concept that makes sense when discussing single trajectories of systems — only the macroscopic view of many copies of that system evolving according to the same dynamics.