Challenging behaviors significantly impact learning and socialization of autistic children and can stress and burden their caregivers. Documentation of challenging behaviors is fundamental for identifying what environmental factors influence them, such as how others respond to a child's such behaviors. Caregiver-tracked data on their child's challenging behaviors can help clinical experts make informed recommendations about how to manage such behaviors. To support caregivers in recording their children's challenging behaviors, we developed GeniAuti, a mobile-based data-collection tool built upon a clinical data collection form to document challenging behaviors and other clinically relevant contextual information such as place, duration, intensity, and what triggers such behaviors. Through an open-ended deployment with 19 parent-child pairs and three expert collaborators, caregivers found GeniAuti valuable for (1) becoming more attentive and reflective to behavioral contexts, including their own response strategies, (2) discovering positive aspects of their children's behaviors, and (3) promoting collaboration with clinical experts around the caregiver-tracked data to develop tailored intervention strategies for their children. However, participant experiences surface challenges of logging behaviors in social circumstances, conflicting views between caregivers and clinical experts around the structured recording process, and emotional struggles resulting from recording and reflecting on intensely negative experiences. Considering the complex nature of caregiver-based health tracking and caregiver--clinician collaboration, we suggest design opportunities for facilitating negotiations between caregivers and clinicians and accounting for caregivers' emotional needs.