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A representation that accurately predicts a deformable object's response is not necessarily the one most useful for choosing manipulation actions. We study this distinction between predictive utility and decision utility in planar single-grasp shape control of Deformable Linear Objects (DLOs). As a common decision backbone, we introduce Geometry-Pointer-Flow (GPF), a geometry-grounded policy that separates grasp-location selection from grasp-conditioned motion generation. Its geometry-only variant reduces one-step executed shape error by 32.8% relative to a DLO-specific diffusion policy, while using 18.7 times fewer parameters and achieving approximately 193 times faster inference. On this backbone, we compare physical context represented by ground-truth parameters, estimated parameters, and learned interaction-history representations, for both forward prediction and executed control. For prediction, ground-truth physical parameters reduce next-shape error by 15.8% over geometry alone, and estimated parameters are slightly more predictive than interaction history. This ordering does not carry over to control: estimated parameters provide no detectable one-step control gain, whereas history-conditioned GPF reduces executed error by 9.9% relative to geometry-only GPF across 470 paired simulation trials. In closed loop, interaction history improves strict-threshold attainment in simulation and yields the strongest zero-shot performance on hardware. All policies are trained exclusively in simulation and transferred to the physical robot without real-world training or fine-tuning. These results show that physical context should be evaluated by the decisions it supports rather than by predictive accuracy alone. Code will be released upon acceptance.