In Lorentz-Finsler geometry it is natural to define the Finsler Lagrangian
over a cone (Asanov's approach) or over the whole slit tangent bundle (Beem's
approach). In the former case one might want to add differentiability
conditions at the boundary of the (timelike) cone in order to retain the usual
definition of lightlike geodesics. It is shown here that if this is done then
the two theories coincide, namely the `conic' Finsler Lagrangian is the
restriction of a slit tangent bundle Lagrangian. Since causality theory depends
on curves defined through the future cone, this work establishes the essential
uniqueness of (sufficiently regular) Finsler spacetime theories and Finsler
causality.Comment: 11 pages. v2: shortened introduction, added references. Changed
title. The previous title was: The definition of Finsler spacetim
