I find that, generally, "normal" refers to the idea that you can function in your day-to-day life without having any particular difficulties, brought on by your own psyche. So, for example, depression causes us to have motivational and perspective problems, making it difficult to funtion in daily life- some people who suffer with depression find it difficult to even get out of bed in the morning.
I find that psychology is a growing, learning kind of science. We don't always get things right (Good example: we used to say that homosexuality was a mental illness, and it was in the DSM. It is no longer considered an illness, and has been taken out of The Book), and sometimes, we are being told about a mental illness and think "wow, that sounds like me!" But, the distinction is that we can still function within our daily lives, and find our creative, physical outlets through, or for, these issues.
As a writer, I'm a little sad to read that article- I know that many writers have killed themselves, suffered depression and anxiety... but I think a great many have also lived to have full, glorious lives, too!
I know that I have suffered depression in the past, and I still deal with anxiety... but it's a process, I am learning to get rid of the anxiety.
I also believe that our society, unfortunately, does not appreciate artistic people enough. In Greek times, creative arts were highly regarded, and you could even get grants- people chosen to sponsor a particular festival were greatly rewarded for doing so, and if they refused, were practically shunned from their society! However, if you look at schooling and news, we find that Science, Math and such are very highly regarded, and the Arts... really aren't. It's very unfortunate, and gives us a certain opinion of people who engage and, indeed, cannot STOP engaging in those arts.
So... yes, all of us may identify to some extent with many of the diagnoses of mental illnesses... but, until it becomes detrimental to us, and gets in the way of our ability to live full and happy lives, I wouldn't consider it an "illness." More an interesting way of looking at the world, and some quirk to our personalities that make us who we are!