There is a Dreamliner (the 787), and there is a Dreamlifter (a highly modified 747 used for hauling large cargo, of which something like four have been built):
These two names (Dreamliner/Dreamlifter) are so close that it's not hard to mix them up - the Dreamlifter was built in support of the Dreamliner program, so it was both named and painted to reflect that heritage.
I would suggest that the initial mention by Ground of the "Dreamliner" traffic was actually meant to refer to a Dreamlifter that was on the field at the time. This is also consistent with reference to an enlarged 747, which is a statement of fact - not a joke - intended to assist the pilots of 2928 in identifying their ground traffic.
This also explains the back and forth, attempting to clarify the incorrect "Dreamliner" terminology. So I believe your assumption that the "Dreamlifter" comment was a joke from a comedian is off. I expect that was someone pointing out that the actual aircraft was a Dreamlifter (I haven't heard the audio, but maybe that was even the Dreamlifter crew checking in with the correction, after hearing themselves misidentified?) And upon hearing and understanding the correction, Ground concurs: "Lifter..."
Given that context, it doesn't seem like any of the above was intended as a joke, but like I said, I haven't dug up the audio from the archives.
All of this would also explain the context of the "a little overweight" comment, which clearly
was a joke....
I just checked FlightAware, and one of the Dreamlifters - N780BA - was indeed at KJFK on August 20, departing for KCHS at around 2200 local, so this fits your timeline.