Have to agree with Aerocat. The flight crew's comms were crap. First they threaten to declare an emergency if they don't get their way, then they say they have declared when they clearly hadn't. As others have said, the ATCO has a responsibility to ensure separation and I'm sure the runway heading was just to give him time to clear traffic out of the way for the AA flight. He wasn't being difficult, he was being professional in the situation he found himself in.
The pilots, on the other hand, appear to be having a petulant episode.
Why had he already not diverted if fuel was an issue? If he allowed himself to get suckered into holding while eating into his divert fuel, he DOES need a session or two in the sim.
If you manage your flight properly, there is no such thing as a "fuel emergency" (which I'm pretty sure is not recognised outside the US anyway).
As for the exact wording of the call... irrelevant. Once you have used the words "declare" and "emergency" in the US, you have done all that is required. Get over yourselves, British pedants! Right may be on your side (in Europe, anyway),, but this is New York we are talking about...