The study addresses challenges in high-voltage direct-current (HVDC) grids, particularly in multi-terminal DC (MTDC) systems, when dealing with DC-line faults. Motivated by the technological and environmental advantages of HVDC, it explores protective measures, emphasizing the significance of efficient fault diagnosis and monitoring. Recently, non-unit protection strategies utilizing time-frequency domain (TFD) techniques have gained attention for their effectiveness. The proposed approach introduces an innovative method for fault detection and faulty phase identification based on local measurements. This method integrates the Stockwell transform (ST) with a modified recurrent neural network (RNN) equipped with bidirectional scanning capabilities for fault detection and a novel Teager Kaiser energy operator (TKEO)-based scheme for accurate faulty phase identification. The proposed RNN architecture, incorporating gated recurrent units (GRU), operates bidirectionally to address long-term dependency limitations. Additionally, the precise faulty phase identification scheme, based on the cumulative sum of the Teager Kaiser energy difference, represents a novel advancement. Extensive simulations on a multi-terminal HVDC system demonstrate the proposed methodology's effectiveness in fault detection and faulty phase identification, particularly under noisy conditions. The proposed method achieved high fault detection accuracy: 99.92% accuracy, 99.65% precision, 100% recall, and 99.82% F1-score (no noise); and 99.02% accuracy, 96.37% precision, 99.29% recall, and 97.81% F1-score (20 dB). Additionally, 100% accuracy was achieved for phase identification. Moreover, the impact of sampling frequency and smoothing rectors are also investigated in the proposed work. Compared to previous works, our approach showed a significant improvement in both detection speed and accuracy.