No, the definition is cold, gravitationally interacting matter. It just so happens that adding that one extra ingredient reconciles a whole number of different

No, the definition is cold, gravitationally interacting matter. It just so happens that adding that one extra ingredient reconciles a whole number of different observations with theory. That didn't have to be the case. For example, if the dark matter is hot, structure formation doesn't match observations, or if dark matter is composed of massive, compact objects, we would observe a lot more microlensing events. But putting in just the right amount of cold dark matter makes everything we can test so far work. That could be a giant coincidence, but most cosmologists think it's because cold dark matter really exists.