Fast and accurate short-circuit (SC) detection is a prerequisite for ensuring stable and safe operation of the silicon carbide (SiC) Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFETs). This paper proposes a composite SC detection circuit which exploits the positive drain-source voltage oscillation during the static on-state to detect the hard switching faults (HSF), and utilizes the negative one during the turn-on transient to identify the faults under loads (FUL), which exhibiting a broad applicability for different types of SC faults. Additionally, unlike the conventional SC detectors relying on the slope rate of the drain current or the magnitude of the drain-source voltage, the proposed detector is package-agnostic for the monitored SiC MOSFETs, and only requires passive sensing network recalibration when adapting to different device specifications. The effectiveness of the proposed detector is experimentally validated on a SC test platform based on a 1200 V/10 A SiC MOSFET, demonstrating fault detection delays of 130 ns and 41 ns under HSF and FUL conditions, respectively.