Click models are a well-established for modeling user interactions with web interfaces. Previous work has mainly focused on traditional single-list web search settings; this includes existing surveys that introduced categorizations based on the first generation of probabilistic graphical model (PGM) click models that have become standard. However, these categorizations have become outdated, as their conceptualizations are unable to meaningfully compare PGM with neural network (NN) click models nor generalize to newer interfaces, such as carousel interfaces. We argue that this outdated view fails to adequately explain the fundamentals of click model designs, thus hindering the development of novel click models. This work reconsiders what should be the fundamental concepts in click model design, grounding them - unlike previous approaches - in their mathematical properties. We propose three fundamental key-design choices that explain what statistical patterns a click model can capture, and thus indirectly, what user behaviors they can capture. Based on these choices, we create a novel click model taxonomy that allows a meaningful comparison of all existing click models; this is the first taxonomy of single-list, grid and carousel click models that includes PGMs and NNs. Finally, we show how our conceptualization provides a foundation for future click model design by an example derivation of a novel design for carousel interfaces.