The decoding performance of product codes and staircase codes based on iterative bounded-distance decoding (iBDD) can be improved with the aid of a moderate amount of soft information, maintaining a low decoding complexity. One promising approach is error-and-erasure (EaE) decoding, whose performance can be reliably estimated with density evolution (DE). However, the extrinsic message passing (EMP) decoder required by the DE analysis entails a much higher complexity than the simple intrinsic message passing (IMP) decoder. In this paper, we simplify the EMP decoding algorithm for the EaE channel for two commonly-used EaE decoders by deriving the EMP decoding results from the IMP decoder output and some additional logical operations based on the algebraic structure of the component codes and the EaE decoding rule. Simulation results show that the number of BDD steps is reduced to being comparable with IMP. Furthermore, we propose a heuristic modification of the EMP decoder that reduces the complexity further. In numerical simulations, the decoding performance of the modified decoder yields up to 0.2 dB improvement compared to standard EMP decoding.