This paper presents a comprehensive analysis of the information-energy capacity region for simultaneous lightwave information and power transfer (SLIPT) systems over lognormal fading channels. Unlike conventional studies that primarily focus on additive white Gaussian noise channels, we study the complex impact of lognormal fading, which is prevalent in optical wireless communication systems such as underwater and atmospheric channels. By applying the Smith's framework for these channels, we demonstrate that the optimal input distribution is discrete, characterized by a finite number of mass points. We further investigate the properties of these mass points, especially at the transition points, to reveal critical insights into the rate-power trade-off inherent in SLIPT systems. Additionally, we introduce a novel cooperative information-energy capacity learning framework, leveraging generative adversarial networks, to effectively estimate and optimize the information-energy capacity region under practical constraints. Numerical results validate our theoretical findings, illustrating the significant influence of channel fading on system performance. The insights and methodologies presented in this work provide a solid foundation for the design and optimization of future SLIPT systems operating in challenging environments.