Grid-connected electrolyzers with intermediate hydrogen storage offer significant potential for reducing electricity costs through flexible operation under dynamic pricing. A threshold-based scheduling optimization approach is developed that derives interpretable on/off production rules from electricity price signals. The method identifies local price thresholds separating high-price from low-price periods, yielding binary production schedules. Adaptive horizon partitioning-subdividing the scheduling horizon when constant thresholds become infeasible-is combined with a receding horizon strategy that implements only a portion of each optimized schedule before re-optimization. This procedure enables systematic investigation of how characteristics of Integrated Electrolyzer-Storage Systems (IESS) influence cost reduction potential while maintaining computational tractability for both offline analysis and online implementation. A case study applying the approach to historical German electricity prices across 2, 688 scenarios reveals that storage capacity is the primary determinant of cost-saving potential, while start-up and shut-down times within the tested range (up to 120 minutes) show negligible impact. Systems with storage capacities allowing for hydrogen production stops of up to 1.4 days achieved electricity cost reductions of up to 80% compared to steady-state production. Production cycles in optimal schedules predominantly follow 24-hour rhythms, with approximately 30 cycles per month observed for high-performing configurations. These findings indicate that conventional flexibility metrics-focused on ramp rates and start-up times-are inadequate for assessing the cost-saving potential of flexibly operated, grid-connected electrolyzers under dynamic pricing. Instead, equipment wear from the frequent production cycling required by optimal schedules emerges as a key consideration for economic operation.