I am having a hard time understanding the term "heat rejection". I always envision that the heat is radiating and do not understand the term. Why not "heat ejection"?
Leaving aside your numbering of your definitions of the word, and how "useless" you think heat rejection is..Grammatically, the word "reject" is probably preferrable because of the implication that something rejected is useless (definition #4). However, while basic thermo, heat of rejection is really useless, when you get into more complex applications, even heat of rejection can be reused.
I'm not sure what you mean by spontaneous. All cycles require an input of work or heat, otherwies there would be no need for them (that's why making cold water in the winter is called "free cooling" - you use only a cooling tower, not a chiller). The point, though, is that if you are using a heat engine to generate mechanical work, you utilize as much of the heat energy as you can and whatever is left gets ejected/rejected. For that application, I don't know that either term is really better.Surely heat ejection would imply that the heat is (spontaneously) ejected out from within the material (presumably cooling it down). It sounds like something you might design an engine to do.