People increasingly use social media to report emergencies, seek help or share information during disasters, which makes social networks an important tool for disaster management. To meet these time-critical needs, we present a weakly supervised approach for rapidly building high-quality classifiers that label each individual Twitter message with fine-grained event categories. Most importantly, we propose a novel method to create high-quality labeled data in a timely manner that automatically clusters tweets containing an event keyword and asks a domain expert to disambiguate event word senses and label clusters quickly. In addition, to process extremely noisy and often rather short user-generated messages, we enrich tweet representations using preceding context tweets and reply tweets in building event recognition classifiers. The evaluation on two hurricanes, Harvey and Florence, shows that using only 1-2 person-hours of human supervision, the rapidly trained weakly supervised classifiers outperform supervised classifiers trained using more than ten thousand annotated tweets created in over 50 person-hours.