The boundaries of existing explainable artificial intelligence (XAI) algorithms are confined to problems grounded in technical users' demand for explainability. This research paradigm disproportionately ignores the larger group of non-technical end users of XAI, who do not have technical knowledge but need explanations in their AI-assisted critical decisions. Lacking explainability-focused functional support for end users may hinder the safe and responsible use of AI in high-stakes domains, such as healthcare, criminal justice, finance, and autonomous driving systems. In this work, we explore how designing XAI tailored to end users' critical tasks inspires the framing of new technical problems. To elicit users' interpretations and requirements for XAI algorithms, we first identify eight explanation forms as the communication tool between AI researchers and end users, such as explaining using features, examples, or rules. Using the explanation forms, we then conduct a user study with 32 layperson participants in the context of achieving different explanation goals (such as verifying AI decisions, and improving user's predicted outcomes) in four critical tasks. Based on the user study findings, we identify and formulate novel XAI technical problems, and propose an evaluation metric verifiability based on users' explanation goal of verifying AI decisions. Our work shows that grounding the technical problem in end users' use of XAI can inspire new research questions. Such end-user-inspired research questions have the potential to promote social good by democratizing AI and ensuring the responsible use of AI in critical domains.