Recent work in Natural Language Processing has increasingly focused on detecting suicidal intent in textual data, where the main aim is to detect expressions in a binary setting. However, previous research has shown that search results and other mentions of suicide online are not only limited to expressions of suicidal intent. Therefore, previously proposed algorithms and datasets might for example struggle to distinguish between suicidal intent of a user and suicide mentioned in a humorous context. In this article we introduce a new dataset called TWISCO, which proposes an alternative approach to classifying expressions of suicidality online. For this, we use a coding framework developed in Psychology to distinguish between different mentions of suicide. Next, we present a variety of machine and deep learning baselines in three different classification settings (text only, features only and text and features). Furthermore, we introduce a Feature GCN that improves performance over the GCN baseline. Finally, we investigate the hypothesis that feelings of dominance are correlated with users expressing their own suicidality. We provide an in-depth discussion of the trade-offs in classifying suicidal intent online.