Visible crew/equipment: During the final fight, a steady cam operator with a white shirt and black straps around his body is seen during several wide angles. This is not a TV camera, it is a film crewmember. The Director even admits to it in the audio commentary on the DVD.
Revealing mistake: Rocky says "cut my eye" during the fight scene, if you watch closly you can see the man actually squeeze a tube full of red stuff inches away from his eye. It's really quite funny... (01:49:20)
Answer: They mean the entire year, 1976, is America's birth "day," being the bicentennial of the Declaration of Independence. So the fight, taking place on January 1, is the "first sporting event on our two hundredth birthday."