The key difference between blockchain network types lay in the degree of decentralization in administration.
Public (or "permissionless") blockchain networks are open to everyone, and anyone who joins can read or write to the network pseudonymously. All transactions can be traced or read, implying a high degree of trust. Data on public blockchains is secured cryptographically and is immutable.
Private (or "permissioned") blockchains are closed networks run by a central authority who determines who can read, write and participate in consensus (block production). Because private blockchains can offer more customizability, they are useful for sandbox environments. Members of private blockchains often are required to pass KYC authentication, and cannot remain anonymous.
Semi-private blockchains are run by a single entity that allows participation in consensus and access only to users who satisfy their set requirements. This type of blockchain network appeals to business-to-business and government use cases.
Consortiums blockchains reach consensus through a predefined, select group of individuals or nodes. Read and write access may be public or restricted to individual participants. Consortium blockchains are considered "permissioned" and appeal to institutional and enterprise use cases.
The Injective Chain is a public blockchain network, allowing any individual or entity to participate in consensus in a fully permissionless fashion. Given the myriad of centralized layer-2 scaling solutions in the blockchain industry today, it's important to make this distinction.