I keep hearing the South Carolina version and let me tell you, cramming five syllables into a three syllable spot is just distractingly awkward.
The article says that “there are 8 versions so far including New York, New Jersey, Minnesota, Florida, Texas, Cleveland, and California.” Note the “so far”, and also that there is no mention in that list of the New Mexico version that the OP refers to. I’d conclude that additional versions have been released since that article was published; presumably, L.G. is releasing them one at a time. Probably because she’s just one person. 
The customized version of We Built This City that I remember had “New York City” dubbed into the chorus (I think) in the most incredibly amateurish way possible. I found it unbelievable that it was played on the radio.
Yes, here we hear it as “Alabama Alabama I love you!” and after reading the lyrics my friend called me to ask if se was having an auditory hallucination.
Since I heard the Alabama version first (and most often) it sounds jarring when I hear another version.
I’ve heard the “Cleveland” version of the song on two local stations already.
Something similar was done with Terri Clark’s “I Wanna Do It All”: I Wanna Do It All - Wikipedia