Transient stability (TS) and short-term voltage stability (STVS) assessment are of fundamental importance for the operation security of power systems. Both phenomena can be mutually influenced in weak power systems due to the proliferation of power electronic interface devices and the phase-out of conventional heavy machines (e.g., thermal power plants). There is little research on the assessment of both types of stability together, despite the fact that they develop over the same short-term period, and that they can have a major influence on the overall transient performance driven by large electrical disturbances (e.g., short circuits). This work addresses this open research challenge by proposing a methodology for the joint assessment of TS and STVS. The methodology aims at estimating the resulting short-term stability state (STSS) in stable, or unstable conditions, following critical events, such as the synchronism loss of synchronous generators (SG) or the stalling of induction motors (IM). The estimations capture the mechanisms responsible for the degradations of TS and STVS, respectively. The paper overviews the off-line design of the data-driven STSS classification methodology, which supports the design and training of a hybrid deep neural network RCNN (recurrent convolutional neural network). The RCNN can automatically capture spatial and temporal features from the power system through a time series of selected physical variables, which results in a high estimation degree for STSS in real-time applications. The methodology is tested on the New England 39-bus system, where the results demonstrate the superiority of the proposed methodology over other traditional and deep learning-based methodologies. For reference purposes, the numerical tests also illustrate the classification performance in special situations, when the training is performed by exclusively using measurements from generation and motor load buses, which constitute locations where the investigated stability can be observed.