Patrick,
I have no dog in this hunt: it really does not matter to me what the "correct" spelling of his name was. I have never written about Hachen/Haga and probably never will. Almost no eighteenth-century speaker had never seen his name "in print," so given the record that you carefully lay out I would say that most people heard his name as "Haga" and used that in tax lists, censuses, etc. It would be interesting to know whether these documents were produced by English- or German-speakers, but I don't think it matters. It seems clear that most people heard his name as "Haga."
It is equally clear that he signed his own name as "Hachen" three times. I did not find the earlier two instances (one from 1757, the other from a list of church officers [I'm not sure of the year]) and so do not feel authorized to post images those documents, but the signatures are very similar--and they match, as I've said, the handwriting on the back of that paper that Peter Gonter had (a letter written to him by his father-in-law: Hachen) and used as a receipt. To me all this seems conclusive.
I agree with you that it is odd that the will, which seems to use the name "Hachen" in the text, uses "Haga" in the signature part. I cannot tell whether the signature is in the same handwriting as the document itself. (Eric suggested that it was.) The "H"s certainly look similar. I wonder if one of the witnesses or whomever Hachen was with at the time signed for him--and, if so, this friend also wrote his name as "Haga" rather than Hachen.
And then there's the gravestone, which is not "Haga" or "Hachen."
In my experience of reading documents nearly every day involving Germans in eighteenth-century Pennsylvania, there are lots and lots and lots of instances of individuals who signed their name one way but whose name is spelled differently in other documents, especially those by English-speakers. It is not uncommon. William Henry writes Ettwein as "Edwine," in Moravian records Dickert is often "Tickert" or other variants. "Haga," "Haggan (the 1752 newspaper spelling), "Hagen" (the gravestone), and "Hachen" may have all sounded about the same, especially when you think how "ch" is pronounced by Germans and how easily the "n" can disappear in speech.
My practice in such situations is not to try to resolve something that is not resolvable but rather to present the problem as it is. I think we have a person who, at times, referred to himself as "Hachen" but who was called "Haga" by many others, perhaps even those who knew him well. I myself would probably use "Hachen," since it is the only way we know he signed his own name--but "Haga" may very well have captured better how he said it aloud.