In this article we revisit from a contemporary perspective a classic problem of polaron theory in one space dimension following the variational approach originally taken by Toyozawa. Polaron structure is represented by variational surfaces giving the optimal values of the complete set of exciton and phonon amplitudes for every value of the joint exciton-phonon crystal momentum κ. Through this exfoliation of the exciton-phonon correlations comprising the polaron, characteristic small polaron, large polaron, and nearly free phonon structures are identified, and the manner in which these compete and/or coexist is examined in detail. Through such examination, the parameter space of the problem is mapped, with particular attention given to problematic areas such as the highly quantum mechanical weak-coupling regime, the highly nonlinear intermediate-coupling regime, and to the self-trapping transition that may be said to mark the onset of the strong-coupling regime. Through such examination of the complete parameter space at all κ, it is found that the common notion of a self-trapping phenomenon associated with κ=0 is a limiting aspect of a more general finite-κ phenomenon. Quantities such as phonon number distributions, complete ground state energy bands, and effective masses are obtained for all κ. The inverse problem of associating localized functions with the variational energy bands is addressed, with attention given to the concept of solitons and with the explicit construction of polaron Wannier states. The successes and failures of the Toyozawa method are assessed.