Whose History? History of ideas: History of Being, History of Freedom
History of a particular object: History of The Cross, History of Aristotle's Corpus,
History of social movements: Labor, Race, and Gender struggles
And definitely yes on history of history. That's historiography, in a remarkably consistent if somewhat funny discipline :).
Also, World History, whether we believe in it or no, is at least a thing. And cliometric history, and all the various ways of writing a history of [x] should probably be included.
I don't think that uses of "History" are univocal. I can see overlapping between various uses, but I wouldn't say that they are univocal at all.
It seems to me that History is about what we care about. We explore history to find reasons to justify our actions, to find our identities, to think through problems, or to understand some topic or other.