Swimming microorganisms such as flagellated bacteria and sperm cells have fascinating locomotion capabilities. Inspired by their natural motion, there is an ongoing effort to develop artificial robotic nanoswimmers for potential in-body biomedical applications. A leading method for actuation of nanoswimmers is by applying a time-varying external magnetic field. Such systems have rich and nonlinear dynamics that call for simple fundamental models. A previous work studied forward motion of a simple two-link model with a passive elastic joint, assuming small-amplitude planar oscillations of the magnetic field about a constant direction. In this work, we found that there exists a faster, backward motion of the swimmer with very rich dynamics. By relaxing the small-amplitude assumption, we analyze the multiplicity of periodic solutions, as well as their bifurcations, symmetry breaking, and stability transitions. We have also found that the net displacement and/or mean swimming speed are maximized for optimal choices of various parameters. Asymptotic calculations are performed for the bifurcation condition and the swimmer's mean speed. The results may enable significantly improving the design aspects of magnetically actuated robotic microswimmers.