FWIW, I think the term itself does potentially carry the broader definition, but like most words, the actual meaning varies in the precise context. I favour the definition over at dictionary.com, to wit, "pity for oneself, esp. a self-indulgent attitude concerning one's own difficulties, hardships, etc."
In other words, I think self-pity is a term used to convey the speaker's opinion that the feelings/behaviour complained of have limited basis or are expressed in an exaggerated manner.