I don't understand your "other hand." Using your logic, not only are
peers and customers indistinguishable, but peers and suppliers are
also indistinguishable.
A customer, on the other hand, is someone who receives your bill, even
if that bill is hugely discounted, with the expectation that if it is
not paid, the connectivity received as a service will be terminated.
A peer is simply a supplier who gives a limited selection of connectivity,
usually constrained by BGP announcement filtering, for a heavy discount,
sometimes as much as 100% off normal pricing.
The problem with equating "peer" with either customer or supplier is
you can never figure out who is the customer and who is the supplier.
If I send you a bill every month for $0.00, does that make me the supplier
and you the customer?
To make things even more confusing, there are usually additional requirements
peers must comply with, that neither customers nor suppliers do. Why do you
think the lawyers go cross-eyed everytime they try to understand a peering
agreement.