Some default library names MC uses:
Read-Only Library: a temporary library MC creates when connecting to another already-open library (local disk or via network FS)
Connected Library: the library created / used when MC connects to another Library Server
Library: the name MC uses for its default Main library
I've never seen the Connected Library (Local) instance, and wasn't able to have MC create, so I'm not sure what configuration or events trigger this name.
Some folks here are use the term "local" too loosely, with respect to a library, and this is causing confusion. MC always operates against a library folder structure that resides somewhere on the host's available file systems. That library folder may reside on a local disk, or reside on a remote file system which is accessible by the host.
The important point regarding performance is to have that library folder on a directly attached disk, not on a remote file system.
Regarding DLNA; MC will create a library folder filled with references to content on a DLNA device if you select and load it. This is for performance, so that MC can act on the cached data, and so that the MC internals just work like any other situation, since MC internally works with the cached metadata and not directly on files (until required).