No, you're a towel!
First of all let me just say. Anybody that doesn't support their home nation is a douche.
Now, I think your interpretation of Support and Support and the correlation between them is different to mine. I am a Juventus supporter, I am a Fingal supporter, I am an éire supporter. I fail to see how I can lend my support towards a team that are in a separate big competition. We support Inter in the CL to keep the coefficient high, we aren't fully-fledged Merda-fags similarly I support North Korea in their quest for world cup glory. It's temporary support not lifetime support, and that doesn't make you a fake fan.
We do indeed have different interpretations of support. I believe that if you want to see the underdogs win against the big, well-"supported" (har har) sides, that doesn't make you a fan or supporter whether it be in the long run or short run because apart from that match you most likely don't care about the said team. The only reason why you want them to beat their opponent is because you want to see the other side lose. In my view, that's not supporting or being a fan of.
Just because I want to see Lyon thrash Manure doesn't mean I'm a supporter of the former.
What makes folks faux fans is actually trying to support the likes of Italy.
You do know that it can easily be said why just nations? Why not towns or cities that are local to you. Why are you guys supporting a Turin based team when your on the other side of the world and have no home affiliation to this city or country for that matter.
It's much, much different. I don't need Italian citizenship to play for Juventus, but I would need citizenship to play for the Azzurri.
That's where the line should be drawn. If not, then international football isn't worth anything and one might as well do away with it.
I support the US in every match they play even against the Azzurri, I agree Italiens are gay boys, and I regularly rip on the Azzurri and their greasy mops. I would not consider myself an "Azzurri fan" but I like the team and support them when it does not harm my own country.
But once the United States goes out I am going to openly support the Azzurri for this tournament but I will not refer to them as "us". Am I a fake US fan?
I'd say yes, but you'd be more of a fake Italian fan.
There's absolutely no reason at all why you need to support the Azzurri. It's just all phony bullshit masked as gloryhunting.