  We revisit the physical arguments which lead to the definition of the
stress-energy tensor $T$ in the Lorentz-Finsler setting $(M,L)$ starting at
classical Relativity. Both the standard heuristic approach using fluids and the
Lagrangian one are taken into account. In particular, we argue that the
Finslerian breaking of Lorentz symmetry makes $T$ an anisotropic 2-tensor (i.
e., a tensor for each $L$-timelike direction), in contrast with the
energy-momentum vectors defined on $M$. Such a tensor is compared with
different ones obtained by using a Lagrangian approach. The notion of
divergence is revised from a geometric viewpoint and, then, the conservation
laws of $T$ for each observer field are revisited. We introduce a natural {\em
anisotropic Lie bracket derivation}, which leads to a divergence obtained from
the volume element and the non-linear connection associated with $L$ alone. The
computation of this divergence selects the Chern anisotropic connection, thus
giving a geometric interpretation to previous choices in the literature.
