Peering is when you send packets from your network (or your customers' networks) to my network (or my customers' networks). Transit is when you send packets from your network (or your customers' networks) to my network, and I agree to deliver it, regardless of if it's within my network, my customer's networks or beyond my network (via peering or my own transit connections).

Peering is historically settlement free at public exchanges and in private connections with balanced traffic. When a private peering is not balanced, historically the peering would be disconnected, but in the mid 2000's, paid peering started showing up more frequently as an option. Paid peering is often less expensive than transit, but it depends on the networks involved and the volume.

Transit - I send you data and you deliver it to another ISP or backbone provider.

Not transit - I send you data and you deliver it to someone in your own network.

If it were considered "transit" in the same sense as delivering packets from one network to another across large distances, then consumer Internet access should be free because it's always biased in favor of receiving rather than transmitting bytes. Verizon should even be paying its customers to accept the extra bits they get.

It seems that the peering arrangements typically pay for transfer through one network to another, not to get data to the end consumer.

I don't know enough about the issue to say whether that's true or not, but that seems to be the argument they're making.

ISPs pay for transit, tier 1 networks do not (by definition). Level 3 is definitely a tier 1 network, but Verizon is acting as both a tier 1 and an ISP. It isn't a very complex problem, tier 1 networks don't pay transit.