In this paper, we consider the convertible codes with the maximum distance separable (MDS) property, which can adjust the code rate according to the failure rates of devices. We first extend the notion of convertible codes to allow initial and final codes with different parameters. Then, we investigate the relationship between these parameters and thus establish new lower bounds on the access cost in the merge and split regimes. To gain a deeper understanding of access-optimal MDS convertible codes in the merge regime, we characterize them from the perspective of parity check matrices. Consequently, we present a necessary and sufficient condition for the access-optimal MDS convertible code in the merge regime. Finally, as an application of our characterization, we construct MDS convertible codes in the merge regime with optimal access cost based on the extended generalized Reed-Solomon codes.