The COVID-19 pandemic required educators and learners to shift to emergency remote instruction, often with little prior notice. To understand how teachers managed the transition, from April to September 2020 we surveyed nearly 1,500 instructors from 118 countries. Using cluster analysis, we have detected two readily distinguishable groups of instructors: a group who were more "engaged" in remote instruction and "coping" with the challenges of online teaching more successfully, and another group who scored lower on both of these fronts. We compare the two groups in terms of their sociodemographic characteristics, and assess the relationship between each sociodemographic marker and teachers' engagement and coping. Overall, our results suggest that teachers were most engaged and coped best with the transition when they had prior experience with remote instruction, worked in the higher education sector, and used real-time synchronous modalities. We also find non-trivial results regarding teachers' gender, years of teaching experience, and their country's level of economic development, while observing no relationship between teachers' age and their levels of engagement or coping. The detection of the contextual effects underscores the importance of large multisite research.