I just realized I was using $\lambda$ for two different things. Maybe the question makes more sense now.

Maybe I should clarify: Calabi's trick is to move the basepoint a little so the distance function becomes smooth again. But you need the old maximum to stay a maximum, which only works when $\gamma<0$. This is not an issue when local uniform ellipticity is assumed.

@DeaneYang Calabi only considers locally uniformly elliptic operators and essentially repeats the usual proof of the strong maximum principle.

For future readers, this question has been answered affirmatively by Bamler and Kleiner arxiv.org/abs/1709.04122.

Although it's not hard to extend the maximum principle for all time by just taking $M\times [0,T]$ and letting $T\to\infty$...compactness of the spatial part is the crucial thing. Though in Chow's book there are maximum principles for complete noncompact manifolds.

Lieberman works on spacetime domains in $\Bbb R^{n+1}$, sure. My claim was not that you can quote his result, my claim is that the proof works for the more general case. What you need, exactly, is for $\Omega$ to have compact closure in $M\times [0,\infty)$. If $M$ is compact and you're only interested in finite times, then you get this for free, of course.

In my answer I gave a short proof of the parabolic maximum principle for subsolutions of a Dirichlet--Cauchy problem. This is far from being as general as possible. Lieberman has more maximum principles, which, as a rule of thumb, always carry over to manifolds whenever the manifold in question is compact. For more maximum principles, see Protter--Weinberger or Volume 2 of Chow et at.'s Ricci flow saga.