To me, the defining characteristic of a sense is that it assimilates information from the external environment for your body to use if necessary. This is also the way the textbooks I've taught from treat it. Thermoception senses external temperature, your body evaluates and, if need be, you take action. Equilibroception senses gravity to help you evaluate your orientation with respect to the Earth.
Proprioception is a hard one to categorize. Some of what is classified as such seem to rely on only internal sensors such as how much a muscle is stretched. Other aspects of proprioception rely on muscle tension which would indeed be affected by gravity and qualify proprioception as a sense by this definition.
I would say fatigue is not a sense because it is based on internal messages. How about a "sense of hunger"? I'd again say no, hunger is also the result of internal messages. Your sense of smell can lead to hunger but the information gathered was odor, not hunger.