Saying that you are "from the country" is not the same as saying that you "are country." One is about where you are from, where you grew up, and the other is about passing some litmus test. I grew up in a rural area, but would never pass such a litmus test.
Here are the lyrics to the Brantley Gilbert song. It's not about passing a litmus test. It's a song about pride:
In every state, there's a station
Playin' Cash, Hank, Willie, and Waylon
In foreign cars and four-wheel drives
There's cowboys and hillbillies
From farm towns to big cities
There ain't no doubt in my mind
Country must be country wide
Here are the Tracy Byrd lyrics. Same thing:
Way back up in the country, back in the hills
Down in the hollows where the folks are real
Livin with the crazies and the old wildcats
Sawed off shotguns and coonskin caps
Thats where Im from and Im proud to say
Im from the country and I like it that way