A variety of analytical and computational models have been proposed to answer the question of why some protein structures are more "designable" (i.e., have more sequences folding into them) than others. One class of analytical and statistical-mechanical models has approached the designability problem from a thermodynamic viewpoint. These models highlighted specific structural features important for increased designability. Furthermore, designability was shown to be inherently related to thermodynamically relevant energetic measures of protein folding, such as the foldability F and energy gap Delta(10). However, many of these models have been done within a very narrow focus: Namely, pair-contact interactions and two-letter amino-acid alphabets. Recently, two-letter amino-acid alphabets for pair-contact models have been shown to contain designability artifacts which disappear for larger-letter amino-acid alphabets. In addition, a solvation model was demonstrated to give identical designability results to previous two-letter amino-acid alphabet pair-contact models. In light of these discordant results, this report synthesizes a broad consensus regarding the relationship between specific structural features, foldability F, energy gap Delta(10), and structure designability for different energy models (pair-contact vs solvation) across a wide range of amino-acid alphabets. We also propose a novel measure Z(d)(k) which is shown to be well correlated to designability. Finally, we conclusively demonstrate that two-letter amino-acid alphabets for pair-contact models appear to be solvation models in disguise.