Agonism plays a vital role in democratic dialogue by fostering diverse perspectives and robust discussions. Within the realm of online conflict there is another type: hateful antagonism, which undermines constructive dialogue. Detecting conflict online is central to platform moderation and monetization. It is also vital for democratic dialogue, but only when it takes the form of agonism. To model these two types of conflict, we collected Twitter conversations related to trending controversial topics. We introduce a comprehensive annotation schema for labelling different dimensions of conflict in the conversations, such as the source of conflict, the target, and the rhetorical strategies deployed. Using this schema, we annotated approximately 4,000 conversations with multiple labels. We then train both logistic regression and transformer-based models on the dataset, incorporating context from the conversation, including the number of participants and the structure of the interactions. Results show that contextual labels are helpful in identifying conflict and make the models robust to variations in topic. Our research contributes a conceptualization of different dimensions of conflict, a richly annotated dataset, and promising results that can contribute to content moderation.