With the increasing number of renewable generations, the prospects of long-distance bulk power transmission impels the expansion of point-to-point High Voltage Direct Current (HVDC) grid to an emerging Multi-terminal high-voltage Direct Current (MTDC) grid. The DC grid protection with faster selectivity enhances the operational continuity of the MTDC grid. Based on the reactor voltage gradient (RVG), this paper proposes a fast and reliable fault identification technique with precise discrimination of internal and external DC faults. Considering the voltage developed across the modular multilevel converter (MMC) reactor and DC terminal reactor, the RVG is formulated to characterise an internal and external DC fault. With a window of four RVG samples, the fault is detected and discriminated by the proposed main protection scheme amidst a period of five sampling intervals. Depending on the reactor current increment, a backup protection scheme is also proposed to enhance the protection reliability. The performance of the proposed scheme is validated in a four-terminal MTDC grid. The results under meaningful fault events show that the proposed scheme is capable to identify the DC fault within millisecond. Moreover, the evaluation of the protection sensitivity and robustness reveals that the proposed scheme is highly selective for a wide range of fault resistances and locations, higher sampling frequencies, and irrelevant transient events. Furthermore, the comparison results exhibit that the proposed RVG criteria improves the discrimination performance of the protection scheme and thereby, proves to be a better choice for future DC fault identification.