Stress-strength reliability, defined as R=P(Y<X), plays a vital role in evaluating a system's ability to withstand stress, especially in complex engineering scenarios. This study investigated R when both the stress Y and strength X followed independent Weibull distributions with a common shape parameter and distinct scale parameters. The analysis was conducted under an improved adaptive progressive Type-Ⅱ censoring scheme. We employed both classical and Bayesian estimation techniques. Classical inference was performed using maximum likelihood estimation and the maximum product of spacings methods, providing point and interval estimates based on their statistical properties. For Bayesian analysis, we proposed two approaches, one based on the likelihood function and the other on the spacings function, using independent gamma priors. Posterior estimates were obtained via Markov Chain Monte Carlo under a squared error loss, along with corresponding credible intervals. A comprehensive simulation study evaluated and compared the performance of the four estimators, two classical and two Bayesian, across varying censoring scenarios. The proposed methods were further validated using real-world data from organic white light-emitting diode devices, illustrating their practical utility.