Operational prediction of the SYM-H index is critical in space weather forecasting and geomagnetic modeling. Although various approaches have significantly deepened our knowledge of SYM-H and continuously improved prediction accuracy, there are still two outstanding issues that cannot be ignored: the requirement that models remain reliable when the input solar wind parameters are provided by different satellites (i.e., cross-satellite generalization) and the need for a precise metric that can evaluate the effectiveness of incorporating the input solar wind parameters and highlight the model's advantage over the persistence model, which often yields misleadingly good scores under traditional metrics like root mean square error (RMSE) or metrics based on correlation. In this work, we construct a straightforward bidirectional long short-term memory (BiLSTM) neural network. By exclusively training on Advanced Composition Explorer (ACE) observations and testing with input solar wind parameters from both ACE and the Deep Space Climate Observatory (DSCOVR), we demonstrate that this architecture is effective in achieving robust cross-satellite generalization. Specifically, for a 1-hour lead time, the operational BiLSTM model achieves an RMSE of 7.36 nT and an
R^2 
of 0.968 on the ACE test dataset, maintaining comparable accuracy on the DSCOVR dataset (
\rmRMSE=7.83 
nT,
R^2=0.956 
). We also report the statistical performance of the BiLSTM model in predicting the storm peak and propose a novel morphological skill score
S_\rmpeak 
that addresses the metric limitation. The new metric also reveals the underlying imperfections in cross-satellite generalization that are masked by conventional evaluations.