Interest is growing in artificial empathy, but so is confusion about what artificial empathy is or needs to be. This confusion makes it challenging to navigate the technical and ethical issues that accompany empathic AI development. Here, we outline a framework for thinking about empathic AI based on the premise that different constellations of capabilities associated with empathy are important for different empathic AI applications. We describe distinctions of capabilities that we argue belong under the empathy umbrella, and show how three medical empathic AI use cases require different sets of these capabilities. We conclude by discussing why appreciation of the diverse capabilities under the empathy umbrella is important for both AI creators and users.