It seems that the customer in question uses Verizon as an ISP - his computer is hooked up to Verizon's network, he isn't using Verizon servers or storage (this isn't really made clear, although the last quote in the article strongly suggests it). My naive reading of Section 512.a [cornell.edu] seems to say that as long as the ISP isn't hosting the questionable material on their own servers, they are not liable. It doesn't even matter if they have been informed of infringement, provided they transmit data only at the request of the endpoints, and do not modify or filter content.
The RIAA is making their case with 512.c [cornell.edu], which relates to "material that resides on a system or network controlled or operated by or for the service provider". But the nuance is that a computer hooked up to Verizon's network (and thus residing "on" it) is subject to far more draconian regulation than a network hooked up to Verizon's network - which would place Verizon in the Transitory category subject to 512.a.
Am I missing something?