The generalized entropy theory (GET) of polymeric glass-forming liquids is reformulated into a computationally simpler and more natural formalism than the original version of this theory. The new theoretical framework greatly facilitates establishing essential trends in the dependence of the segmental relaxation time τ, fragility, characteristic temperatures of glass-formation, etc., on the combined influences of monomer molecular structure, chain rigidity, and cohesive interaction strength. Special attention is placed on the estimating the parameters of the phenomenological Vogel-Fulcher-Tammann relations for describing segmental relaxation in diverse liquids in the low temperature range of glass-formation, Tg > T > Tc (or Tg < T < Tg + 100 K), where Tg and Tc are, respectively, the glass transition temperature and the crossover temperature separating the high and low temperature regimes of glass-formation. Finally, we discuss how the molecular energetic interaction parameters of the GET can be estimated from experimental data. Illustrative calculations are performed for the stiffness factor σ and the cohesive energy density u as a first step in this direction.