Underwater Wireless Optical Communication is a key enabling technology for future space-air-ground-sea integrated networks. However, UOWC faces critical hurdles from spatial randomness and stringent energy constraints. These challenges fundamentally limit network lifetime and sustainability. This paper develops a comprehensive stochastic geometry framework to perform a differential energy analysis of UOWC links.Instead of relying on simplified models, we employ a three-dimensional truncated Poisson point process to accurately capture the anisotropic nature of the underwater environment, specifically the disparity between horizontal spread and vertical depth. It incorporates a Lambertian emission pattern, random receiver positions and orientations, and a realistic channel model with extinction effects. Under this model, we derive a full suite of closed-form expressions for key performance indicators. These include the nearest-neighbor distance distribution, expected received power, SNR, and BER. A principal and counter-intuitive finding of our analysis is an offset-pointing strategy. This strategy involves intentionally misaligning the receiver by a deterministically optimal angle. This approach maximizes the integrated received power across the aperture, contrary to the conventional pursuit of perfect alignment. We formulate and solve an energy-efficiency optimization problem. Our results demonstrate that this strategy enhances system robustness and yields substantial performance gains. Simulation results validate our analytical models. They show that the optimal offset strategy can reduce the required transmit power by nearly 20\% to achieve a target BER. This reduction directly translates into extended network lifetime and higher total data throughput. These findings offer a new design paradigm for deploying robust, cost-effective, and sustainable UOWC networks.