Appropriate traffic state representation is crucial for learning traffic signal control policies. However, most of the current traffic state representations are heuristically designed, with insufficient theoretical support. In this paper, we (1) develop a flexible, efficient, and theoretically grounded method, namely generalized phase pressure (G2P) control, which takes only simple lane features into consideration to decide which phase to be actuated; 2) extend the pressure control theory to a general form for multi-homogeneous-lane road networks based on queueing theory; (3) design a new traffic state representation based on the generalized phase state features from G2P control; and 4) develop a reinforcement learning (RL)-based algorithm template named G2P-XLight, and two RL algorithms, G2P-MPLight and G2P-CoLight, by combining the generalized phase state representation with MPLight and CoLight, two well-performed RL methods for learning traffic signal control policies. Extensive experiments conducted on multiple real-world datasets demonstrate that G2P control outperforms the state-of-the-art (SOTA) heuristic method in the transportation field and other recent human-designed heuristic methods; and that the newly proposed G2P-XLight significantly outperforms SOTA learning-based approaches. Our code is available online.