This paper investigates using multi-label deep learning approach to extending the understanding of cognitive presence in MOOC discussions. Previous studies demonstrate the challenges of subjectivity in manual categorisation methods. Training automatic single-label classifiers may preserve this subjectivity. Using a triangulation approach, we developed a multi-label, fine-tuning BERT classifier to analyse cognitive presence to enrich results with state-of-the-art, single-label classifiers. We trained the multi-label classifiers on the MOOC discussion messages that were categorised into the same phase of cognitive presence by the expert coders, and tested the best-performing classifiers on the messages that the coders categorised into different phases. The results suggest that multi-label classifiers slightly outperformed the single-label classifiers, and the multi-label classifiers predicted the discussion messages as either one category or two adjacent categories of cognitive presence. No messages were tagged as non-adjacent categories by the multi-label classifier. This is an improvement compared to manual categorisation by our expert coders, who obtained non-adjacent categories and even three categories of cognitive presence in one message. In addition to the fully correct prediction, parts of messages were partially correctly predicted by the multi-label classifier. We report an in-depth quantitative and qualitative analysis of these messages in the paper. The automatic categorisation results suggest that the multi-label classifiers have the potential to help educators and researchers identify research subjectivity and tolerate the multiplicity in cognitive presence categorisation. This study contributes to extending the literature on understanding cognitive presence in MOOC discussions.