C2S is about interoperability between any kind of Client and any kind of Server. What that looks like in practice is yet to be seen, but it means that yes the standard ActivityPub inbox payload is sent from the Server to Client and it is up to the Client to render each inbox item as it seems appropriate.
This would let any Client “just work” with Pleroma, PeerTube, Funkwhale (if they supported C2S) out of the box.
The C2S protocol basically replaces the REST paradigm and the custom APIs built with it.
Edit: A Client implementation, for example, could decide to use the inbox to construct a list of not-deleted-IRIs of objects, and then make an AP request fetching those objects and rendering them asynchronously.
A different Client implementation could render Activities directly (“Addison created this note: …”) in addition to the above.
A different client could cache and render Objects based only on what is received in Creates in its inbox.
Nothing prevents a client from using specific APIs, but then they’re tied to specific ActivityPub software and are no longer pure C2S.