This two-part paper presents a generic methodology for measuring the short-term voltage stability (STVS) of power systems dominated by inverter-based resources (IBRs), which introduces the concept of generalized voltage damping (GVD) for quantifying STVS from both global and local perspectives. It leads to a model-independent approach to assessing the voltage stability, the system strength, and the capability of dynamic devices to support voltage during transient process. Part I of this paper focuses on deriving the system-wise generalized voltage damping (sGVD) index and its applications. The sGVD index is defined as the decay rate of voltage-related transient energy (VTE) dissipated on the (aggregated) buses of the power system, which can be obtained using the maximum Lyapunov exponent (MLE) technique. The proposed sGVD index is theoretically demonstrated to capture the actual voltage damping of devices and to be strongly linked with STVS. These unique properties enable a model-independent approach to measuring STVS and system strength, even in the presence of heterogeneous and strongly nonlinear dynamics of IBRs. We verify the theoretical results by conducting simulations on the modified IEEE 39-bus system and two large-scale practical power systems with integration of massive renewable resources, demonstrating the effectiveness and practicality of the methodology.