Definition: An Internet (IX) is a layer 2 service to facilitate the interconnection between Internet Service Providers and Large Scale Network Savvy Content Providers.
An IX facilitates peering between players, usually across a public and/or private peering fabric of some type. Ethernet is the most common IX public peering fabric.
Definition: Public Peering is Internet Peering across a shared (more than two party) peering fabric.
Most Internet Exchange Points offer some form of public peering, typically using ethernet as the layer 2 fabric. In some cases, private peering is a reasonable alternative or complementary service.
Most IXes enable or support both public and private peering. See the US vs European Exchange Points for a description of alternative models of IXes.