Close but not quite.
Internet customer: someone who pays me to connect them to "the Internet"
Transit provider: Someone I pay to connect me to "the Internet"
Peer: Any network with whom I agree to trade packets from my own network and my Internet customers, to his network and Internet customers, and vice versa. Put another way: we agree to trade packets only for which our respective customers have paid us to transmit or receive them. Sometimes called "settlement free peering."
You'll sometimes hear the term "paid peering" but that's actually Transit above that the Internet customer chooses to use a particular way.
BGP peer: any connection between Internet networks which uses BGP, whether customer, transit or "peer." Not relevant to this discussion, but I call it out to keep the term from creeping in as if it was the same thing as "peering."
Peering is most definitely a net neutrality issue. When one network refuses to peer with another, they force the other network to either take the long way around (throttling) or to pay for transit service (paid prioritization).
There's room for nuance here. There's no suggestion that one network must peer in exactly the manner another wants to. Or that one network must pay the direct costs of connecting with another just because the other wants to. But a flat refusal to peer based on some arbitrary and capricious measurement of the other network's nature is most emphatically a net neutrality violation.