Neural network-based decoding methods show promise in enhancing error correction performance but face challenges with punctured codes. In particular, existing methods struggle to adapt to variable code rates or meet protocol compatibility requirements. This paper proposes a unified long short-term memory (LSTM)-based neural decoder for punctured convolutional and Turbo codes to address these challenges. The key component of the proposed LSTM-based neural decoder is puncturing-aware embedding, which integrates puncturing patterns directly into the neural network to enable seamless adaptation to different code rates. Moreover, a balanced bit error rate training strategy is designed to ensure the decoder's robustness across various code lengths, rates, and channels. In this way, the protocol compatibility requirement can be realized. Extensive simulations in both additive white Gaussian noise (AWGN) and Rayleigh fading channels demonstrate that the proposed neural decoder outperforms conventional decoding techniques, offering significant improvements in decoding accuracy and robustness.