For future millimeter/submillimeter and terahertz astronomy, kilo-pixel imaging arrays of ultra-sensitive, background-limited detectors are essential. Given their intrinsic frequency-domain multiplexing and straightforward fabrication, superconducting kinetic inductance detectors (KIDs) are a leading candidate for this purpose. Aluminum, which has a long quasiparticle lifetime, is a crucial material for implementing the sensitive element of a KID. A key figure of merit that quantifies detector sensitivity is the noise equivalent power (NEP). This study compares two characterization methods—small-signal analysis and a frequency-shift response model—for the optical responsivity and NEP of an aluminum-based terahertz KID coupled to a cryogenic blackbody. The KID is a lumped-element, high-Q microwave resonator consisting of a tantalum interdigitated capacitor in parallel with an aluminum inductor, with the latter acting as the 15 THz absorber. The small-signal analysis method, which uses phase and amplitude as observables, requires high precision in blackbody temperature control and involves long measurement times. In contrast, the frequency shift response model method, which uses frequency and dissipation as observables, imposes less stringent requirements on thermometer resolution and enables faster measurements. Moreover, it fits the fractional frequency shift response more accurately than linear models. Consequently, it represents an efficient and rapid approach to characterizing the optical responsivity and NEP of KIDs. With this method, a minimum optical frequency NEP of 7.5×10^–18 \mathrmW/\sqrt\textHz and a dissipation NEP of 7.1×10^–18 \mathrmW/\sqrt\mathrmHz are achieved for the terahertz KID at 300 Hz, referenced to absorbed power. Furthermore, the frequency NEP significantly exceeds the dissipation NEP at 1, 10, and 100 Hz, which is attributable to two-level system noise. Our work provides valuable technical guidance for the rapid NEP characterization of high-sensitivity terahertz KIDs in low-temperature measurement applications.