The name Hudson River Derby was coined (or is at least heartily supported) by the Third Rail to semantically support their bogus claim on "New York City proper," and emphasize the New Jersey location of Red Bull Arena.
I know its not the biggest deal, but there's no need for us to concede that point and allow them to set the agenda by propagating a name that through constant repetition in the media will reinforce the only 'advantage' they have over Red Bull.
This team is trying to pretend that they are for New York and Red Bull is for New Jersey, and that's why they support a rivalry name that includes the geographical border between the two.
"The Battle of the Hudson" is the hockey series between the New Jersey Devils and the New York Rangers, so in the New York sporting collective unconscious, this already signifies NY vs NJ. Which is fine if Red Bull is willing to concede their New York affiliation.
The blue team is trying to establish an identity by emphasizing their geography over our history and tradition as the team for the Metro area.
So, anyone is free to use Hudson River Derby as much as they want, but be aware that it's psychological warfare and you are helping the enemy.