This paper generalizes the semi-Markov model for limit order books (LOBs) originally proposed by Swishchuk and Vadori. The key innovation is allowing price changes to be of arbitrary size, not restricted to a single tick, which is motivated by empirical evidence showing that a large fraction of price changes exceed one tick. The authors develop two extensions: a two-state model where price changes can take two values (e.g., +δ and -δ but with variable magnitude) and a more general multi-state model allowing an arbitrary number of price change sizes. For both models, they derive diffusion limits under balanced and unbalanced market conditions, providing explicit formulas for the volatility coefficient in terms of model parameters such as transition probabilities, stationary distributions, and mean sojourn times. The theoretical results are complemented by implementations and numerical tests on real LOB data from multiple stocks (Apple, Amazon, Google, Microsoft, Intel, Cisco, Facebook, etc.). The empirical analysis demonstrates that the generalized models significantly improve the fit between the theoretical diffusion coefficient and the observed 10-minute volatility compared to the original fixed-tick model. The paper also reviews key assumptions of previous models and validates them with new data, confirming the need for non-exponential inter-arrival times and variable price change sizes. Overall, this work provides a more realistic and flexible framework for modeling LOB dynamics and price volatility in high-frequency finance.