I think that if you grasp the distinction between a library server and a media server then it will fall into place.
NB: as far as I know this is accurate but it occurs to me that I haven't verified the network traffic follows the path I believe it does so I could be wrong.
Remember that a library is simply a database, it contains a list of items and item has a bunch of tags that describe that item including 1 or more tags that define the content (e.g. a file, a URL, a playback range, content type etc). This library can be served (by the library server) to a variety of different clients including MC clients, DLNA renderers and the various jriver remotes (panel, jremote, gizmo) and those clients can then play (render in DLNA terms) that content.
I get the impression that this much is quite clear to you.
Now take one of those MC clients and turn on media network so that it can itself serve content to clients that connect directly to this instance. If you ignore the local file option, content will be streamed from the library server to this MC instance (acting as a client to that library server) and then this MC instance will stream the content (acting as a media server to that connected client).
If you now turn on the play local content option, the library server is no longer involved in any content rendering, it just serves the database so the media server client can stream that content directly to clients.
to go blow by blow....
To operate a client, the client must be set up as a Library.
if "as a library" means "as a library server" then no
If JRemote is controlling the client instance of MC, is that instance acting as a renderer with the Library Server (somehow?) pushing media to it?
if the local file option is not available then yes, the server has to stream the content to the client
Is it acting as a Controller and controlling the Library Server and causing the Server to push it media?
I can't think of a reason why this isn't technically possible but I don't believe MC can or does do this
Is it using the Library Server's library and streaming the "local" file to itself to pass on to my system?
if you turn on the local file option, yes
Or is it acting as the server streaming the content to my TV with the Library Server not involved at all?
the library server is involved because without the library server, the client doesn't know what it has available to play.
It occurs to me I don't know for sure to what extent a client caches the library, i.e. how resilient it is the server being unavailable. I believe the answer is "not resilient at all", i.e. it has to talk to the server whenever it plays anything.