  We develop an axiomatic geometric approach and provide an unconventional
review of modified gravity theories, MGTs, with modified dispersion relations,
MDRs, encoding Lorentz invariance violations, LIVs, classical and quantum
random effects, anisotropies etc. There are studied Lorentz-Finsler like
theories elaborated as extensions of general relativity, GR, and quantum
gravity, QG, models and constructed on (co) tangent Lorentz bundles, i.e.
(curved) phase spaces or locally anisotropic spacetimes. An indicator of MDRs
is considered as a functional on various type functions depending on phase
space coordinates and physical constants. It determines respective generating
functions and fundamental physical objects (generalized metrics, connections
and nonholonomic frame structures) for relativistic models of Finsler, Lagrange
and/or Hamilton spaces. We show that there are canonical almost symplectic
differential forms and adapted (non) linear connections which allow us to
formulate equivalent almost K\"{a}hler-Lagrange / - Hamilton geometries. This
way, it is possible to unify geometrically various classes of (non) commutative
MGTs with locally anisotropic gravitational, scalar, non-Abelian gauge field,
and Higgs interactions. We elaborate on theories with Lagrangian densities
containing massive graviton terms and bi-connection and bi-metric modifications
which can be modelled as Finsler-Lagrange-Hamilton geometries. An appendix
contains historical remarks on elaborating Finsler MGTs and a summary of
author's results in twenty directions of research on (non) commutative/
supersymmetric Finsler geometry and gravity; nonholonomic geometric flows,
locally anisotropic superstrings and cosmology, etc.
