We argue that predator-prey dynamics, a cornerstone of ecology, can be driven by insufficiently-explored aspects of predator performance that are inherently prey-dependent: i.e., these have been falsely excluded. Classical -Lotka Volterra based- models tend to only consider prey-dependent ingestion rate. We highlight three other prey-dependent responses and provide empirically-derived functions to describe them. These functions introduce neglected nonlinearities and threshold behaviours into dynamic models leading to unexpected outcomes: specifically, as prey abundance increases predators: 1) become less efficient at using prey; 2) initially allocate resources towards survival and then allocate resources towards reproduction; and 3) are less likely to die. Based on experiments using model-zooplankton, we explore consequences of including these functions in the classical structure and show they alter qualitative and quantitative dynamics of an empirically-informed, generic predator-prey model. Through bifurcation analysis, our revised structure predicts: 1) predator extinctions, where the classical structure allows persistence; 2) predator survival, where the classical structure drives predators towards extinction; and 3) greater stability through smaller amplitude of cycles, relative to the classical structure. Then, by exploring parameter space, we show how these responses alter predictions of predator-prey stability and competition between predators. Based on our results, we suggest that classical assumptions about predator responses to prey abundance should be re-evaluated.