Kind of long and somewhat confusing for me. Not being a local, is this a hyphenated city name or two counties or what? Google results didn't help me quite understand it. Obviously Nashville would be the stronger choice but that's reserved.
Nashville-Davidson (balance) is the name used by the U.S. Census Bureau to designate the portion of Davidson County that does not include satellite cities of Nashville.
I'm from the US and I have not heard of it. What wikipedia is suggesting is that this term is relevant to statisticians and likely only recognizable to locals and I doubt anyone there will address their envelopes Nashville-Davidson, TN. Hopefully a local can chime in and confirm.