Conceptual operationalizations of empathy in NLP are varied, with some having specific behaviors and properties, while others are more abstract. How these variations relate to one another and capture properties of empathy observable in text remains unclear. To provide insight into this, we analyze the transfer performance of empathy models adapted to empathy tasks with different theoretical groundings. We study (1) the dimensionality of empathy definitions, (2) the correspondence between the defined dimensions and measured/observed properties, and (3) the conduciveness of the data to represent them, finding they have a significant impact to performance compared to other transfer setting features. Characterizing the theoretical grounding of empathy tasks as direct, abstract, or adjacent further indicates that tasks that directly predict specified empathy components have higher transferability. Our work provides empirical evidence for the need for precise and multidimensional empathy operationalizations.