Hybrid power plants comprising renewable power sources and electrolyzers are envisioned to play a key role in accelerating the transition towards decarbonization. It is common in the current literature to use simplified operational models for electrolyzers. It is still an open question whether this is a good practice, and if not, when a more detailed operational model is necessary. This paper answers it by assessing the impact of adding different levels of electrolyzer details, i.e., physics and operational constraints, to the optimal dispatch problem of a hybrid power plant in the day-ahead time stage. Our focus lies on the number of operating states (on, off, standby) as well as the number of linearization segments used for approximating the non-linear hydrogen production curve. For that, we develop several mixedinteger linear models, each representing a different level of operational details. We conduct a thorough comparative ex-post performance analysis under different price conditions, wind farm capacities, and minimum hydrogen demand requirements, and discuss under which operational circumstances a detailed model is necessary. In particular, we provide a case under which a simplified model, compared to a detailed one, results in a decrease in profit of 1.8% and hydrogen production of 13.5% over a year. The key lesson learned is that a detailed model potentially earns a higher profit in circumstances under which the electrolyzer operates with partial loading. This could be the case for a certain range of electricity and hydrogen prices, or limited wind power availability. The detailed model also provides a better estimation of true hydrogen production, facilitating the logistics required.