Cosserat rod theory proved efficient modeling performances in robotics, especially in the context of continuum robots, in the past decade. The implementation of such theory is far from being unique and straightforward. We consider the illustrative example of multisegment, general routing tendon actuated continuum robots in their nominal static operating regime. This article details two main approaches based on Cosserat rod modeling, namely, the Newtonian and Lagrangian approaches. We provide a walk-through guide regarding theoretical derivations and numerical implementation of both approaches, together with a proof of equivalence. This comparative study is supplemented with novel contributions and extensions of each approach and in-depth discussion of their performances and applicability, as well as highlighting their special features.