I only scanned it briefly, but I could not find Illinois on that page. I've lived in Illinois all my life, and I can't for the life of me figure out what we call ourselves!
Illini would work too, though spell check doesn't seem to like it. But where I come from, we always just called you FIBs.
This is an interesting topic, and honestly I think more than anything else you are what you think you are. No one would find it weird if I was interested in the Revolutionary War, it created the country I live in. But I have no connection to it, my family didn't arrive for another 70 years. 19th century Irish history? I owe my existence to it. So it shouldn't be unreasonable for that to hold an importance place in my core.
I would never say I'm Irish, though. I'm American, and Irish-American (which honestly is more about the experience of Irish immigrants in America than it is about Ireland, though Ireland obviously plays some part in that).