A unified progressive hybrid censoring scheme is introduced by combining progressive and hybrid plans, allowing tests to be terminated either after a predetermined number of failures or at a fixed time. Both likelihood and Bayesian procedures are developed for estimating the parameter, reliability, and hazard rate of a one-parameter lifetime model when data are generated under this scheme. Maximum likelihood estimates are obtained via the Newton-Raphson algorithm, and asymptotic confidence intervals are constructed using the delta method with the Fisher information matrix. In addition, parametric bootstrap methods are employed for constructing confidence intervals. Within the Bayesian framework, Markov chain Monte Carlo techniques are employed under non-informative and informative independent gamma priors, with computational intractability addressed through the Metropolis-Hastings algorithm. Progressive censoring with binomial random removals has also been considered within this framework to enhance flexibility in test termination and data collection. Extensive Monte Carlo simulations are conducted to compare the efficiency of the likelihood and Bayesian estimators across multiple censoring designs, and the superiority of Bayesian inference with informative priors is demonstrated. The applicability of the proposed estimators is illustrated using three real datasets: tensile strength of polyester fibers, aircraft air-conditioning failures, and ordered failure times. The one-parameter model is further compared with ten standard unit distributions. The censoring framework is successfully applied to these datasets, confirming its practical value in modeling reliability and failure behavior.