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Magnetorheological (MR) fluid (MRF) dampers, serving as fail-safe semi-active devices, exhibit nonlinear hysteresis characteristics, emphasizing the necessity for accurate modeling to formulate effective control strategies in smart systems. This paper introduces a novel stop operator-based Prandtl-Ishlinskii (PI) model, featuring a reduced parameter set (seven), designed to estimate the nonlinear hysteresis properties of a large-scale bypass MRF damper with variable stiffness capabilities under varying applied current. With only seven parameters, the model realizes current, displacement, and rate dependencies. The force-displacement and force-velocity responses of the designed MRF damper were experimentally characterized under broad ranges of applied current (0-2 A), excitation frequency (0.5-4 Hz), and displacement amplitude (1-2.5 mm). A training dataset was subsequently used to develop a novel field-dependent modified PI model, incorporating multiple hysteresis operators with and without a friction element. The proposed model accurately predicted the MRF damper behavior within the training dataset, and its validity was assessed against data from diverse experimental conditions. The PI model with friction element generally outperformed the model without friction when frequency exceeds 0.5 Hz, demonstrating its ability to characterize nonlinear hysteresis force-displacement and force-velocity properties of the MRF damper under the ranges of applied current and excitations considered with reasonable accuracy. Experimental data were also estimated by the Bouc-Wen model, and compared with those obtained via the formulated PI model, affirming the overall superiority of the proposed PI models considering the computational cost, and total number of parameters. Leveraging the simplicity, minimal parameter requirements, and analytic invertibility of PI models, the proposed PI model is considered a superior choice for modeling and subsequently controlling smart structures employing MRF dampers.