The facts are that the New York Times has no idea whether this is anything to be concerned about or not. Partly this is because the newspaper doesn't seem to know what the two companies are actually negotiating about, and partly because the reporter doesn't seem to understand quality of service (QoS). OK, there I go throwing around technical acronyms, but the fact is that asking a carrier to support quality of service packets is a lot different from being the end of net neutrality.
The questions not answered are whether Google is asking for exclusivity in this discussion about priority, or whether it's asking for Verizon to honor its QoS flags on video packets from YouTube. Pretty much anyone who uses an Ethernet network for anything involving audio or video streaming knows what QoS does. It basically ensures that the packets arrive in sequence and that they have a higher priority at the router than packets that don't have such flags set.
Using such quality flags is important to voice communications to a greater extent than even video, because your brain is more sensitive to information discontinuities when you hear things. This is why it's so hard to listen to a cell phone call when the signal is really crummy, and why it's just annoying to watch a video feed that has occasional drop outs.
If that's the case, then what Google is really asking from Verizon is that its routers be set to honor QoS flags. This takes time to implement (although not much else since virtually any router purchased since the Stone Age can be configured to honor QoS), which is probably why Verizon wants Google to pay for it.
But does this really affect net neutrality? There's nothing to indicate that any traffic will be excluded in such an arrangement, there's no information so far that Verizon won't also work with anyone else who wants such treatment. In fact, there's nothing to indicate that Verizon won't simply enable QoS on its routers and honor priority that's set by anyone.
Basically, packet handling is part of network management, and network management is allowed under any definition of net neutrality that I've seen. Unless there's some indication that Verizon is planning to hurt other services, deny access to some websites, or degrade access to other video services, then I'm not sure this is really a net neutrality issue.
But the fact is, right now we don't know what Google and Verizon are talking about, and histrionics about the end of net neutrality are premature. Considering the fact that right now this issue is more speculation than fact, it would seem that the New York Times is crying wolf for the sake of Web traffic. Right now they don't know what's planned (if anything) and neither does anyone else except the two companies involved.