  The general notion of anisotropic connections $\nabla$ is revisited,
including its precise relations with the standard setting of pseudo-Finsler
metrics, i.e., the canonic nonlinear connection and the (linear) Finslerian
connections. In particular, the vertically trivial Finsler connections are
identified canonically with anisotropic connections. So, these connections
provide a simple intrinsic interpretation of a part of any Finsler connection
closer to the Koszul formulation in $M$. Moreover, a new covariant derivative
and parallel transport along curves is introduced, taking first a
self-propagated vector ({\em instantaneous observer}) so that it serves as a
reference for the propagation of the others. The covariant derivative of any
anisotropic tensor is given by the natural derivative of a curve of tensors
obtained by parallel transport along a curve and, in the case of pseudo-Finsler
metrics, this is used to characterize the Levi-Civita--Chern anisotropic
connection as the one that preserves the length of parallely propagated
vectors.
