That's only the first part of the request. The highlighted part of the email says they need to remove all mention of them from the site, and specifically calls about that bit of text as something that needs to be removed.
What I wrote in my comment is in reference to the entire request, including the highlighted part of the email. Asking someone to remove something or cease and desist is not solely the domain of copyright infringement. The only part of the second C&D that deals with copyright infringement is the first sentence, where they state that changing the video status to private is unacceptable.
The other claim they make, starting at "Further, you need to remove any reference..." and ending with "...We made a replica in under an hour" is a claim of trademark, not copyright, infringement, contrary to GGP's belief. Nowhere in the C&D does the NYT assert copyright over the use of the quoted text: they only made a claim of copyright infringement in the first C&D with respect to "any New York Times content", not the text referencing their name.
This situation really isn't that hard to parse and requires no speculation: just read all of what the New York Times sent instead of relying on the OP's misinterpretation that everything the NYT is claiming is all just something absurd about copyright infringement.