I do think in that case that fake is not the right adjective to use. Every definition of fake implies intent to deceive. If you're trying to encompass the meanings you describe above, fake doesn't appear to cover it.
I'm still trying to get my head around what fake news is actually trying to describe. I'd like it to refer to intentional deception, but I think it's being used much more broadly than that. I fear that it's effectively meaningless at this point, other than as a generic insult.