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

On axiomatic formulation of gravity and matter field theories with MDRs and Finsler-Lagrange-Hamilton geometry on (co)tangent Lorentz bundles