This paper introduces a Fault Diagnosis (Detection, Isolation, and Estimation) method using Set-Membership Estimation (SME) designed for a class of nonlinear systems that are linear to the fault parameters. The methodology advances fault diagnosis by continuously evaluating an estimate of the fault parameter and a feasible parameter set where the true fault parameter belongs. Unlike previous SME approaches, in this work, we address nonlinear systems subjected to both input and output uncertainties by utilizing inclusion functions and interval arithmetic. Additionally, we present an approach to outer-approximate the polytopic description of the feasible parameter set by effectively balancing approximation accuracy with computational efficiency resulting in improved fault detectability. Lastly, we introduce adaptive regularization of the parameter estimates to enhance the estimation process when the input-output data are sparse or non-informative, enhancing fault identifiability. We demonstrate the effectiveness of this method in simulations involving an Autonomous Surface Vehicle in both a path-following and a realistic collision avoidance scenario, underscoring its potential to enhance safety and reliability in critical applications.