State-of-the-art (SOTA) urban traffic control increasingly employs Multi-Agent Reinforcement Learning (MARL) to coordinate Traffic Light Controllers (TLCs) and Connected Autonomous Vehicles (CAVs). However, the performance of these systems is fundamentally capped by their hand-crafted, myopic rewards (e.g., intersection pressure), which fail to capture high-level, human-centric goals like safety, flow stability, and comfort. To overcome this limitation, we introduce C2T, a novel framework that learns a common-sense coordination model from traffic-vehicle dynamics. C2T distills "common-sense" knowledge from a Large Language Model (LLM) into a learned intrinsic reward function. This new reward is then used to guide the coordination policy of a cooperative multi-intersection TLC MARL system on CityFlow-based multi-intersection benchmarks. Our framework significantly outperforms strong MARL baselines in traffic efficiency, safety, and an energy-related proxy. We further highlight C2T's flexibility in principle, allowing distinct "efficiency-focused" versus "safety-focused" policies by modifying the LLM prompt.