Modern astronomical time-domain surveys routinely collect multi-band light curves that provide complementary information about the physical processes governing source variability. Although various multi-output Gaussian process (GP) models have been applied in different astronomical contexts, the statistical and astrophysical implications of their dependence structures have received comparatively limited attention. In this work, we examine three existing approaches to modeling cross-band dependence within a common multi-output GP framework and investigate how their underlying assumptions affect statistical inference and astrophysical interpretation. These include covariance-based, stochastic differential equation (SDE)-based, and latent-process approaches, which differ mainly in how cross-band dependence is introduced. The covariance-based approach directly specifies cross-band covariance functions to characterize dependence, whereas the SDE-based approach specifies joint temporal dynamics, with cross-band dependence arising from coupled dynamics or correlated driving processes. The latent-process approach represents dependence through shared stochastic processes and their transformations, which can reflect physical relationships among multiple light curves. We demonstrate the practical implications of these modeling choices through applications to multi-band active galactic nucleus light curves and emission-line reverberation mapping. Rather than advocating a universally preferred approach, this work provides guidance for selecting dependence structures according to scientific objectives and clarifies how these choices influence the statistical characterization and scientific interpretation of multi-band stochastic variability.

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