We present a new general mathematical formulation of the Eikonal equations for ray tracing, which can use any internal magnetic field model. The analytical and tractable expressions are conveniently based on a vector formalism, which is independent of any coordinate system. This formalism is newly introduced in the widely used BAS HOTRAY code and tested for frequencies ranging from 1 to 20 kHz, covering lightning and navy transmitter frequencies. We show and quantify the tremendous role of the magnetic field model, testing from the pure dipole field to the full International Geomagnetic Reference Field (IGRF) model. Most often, the longitudinal drift of the rays computed with different geomagnetic models makes a notable difference among them, being absent for rays computed with a pure dipole. The error in path length of the dipole-based rays can be 100% different than the IGRF-based ray. Dipole-based rays are most often drastically erroneous in ray path or final footprint. Rays computed with IGRF or an eccentric-tilted dipole can often agree within a reasonable error range, but not systematically. We find a major importance of the initial wave normal angle and initial azimuthal angle in ray paths. Our study concludes that major ray tracing study should use a realistic magnetic field model to be predictive. Yet, the method remains to be extended to general external magnetic field and more realistic cold plasma density models.