Consolidate logic for protobuf message handling between ssh and web socket clients (#17185)

This is a refactor to prepare for adding LSP support in SSH remote
projects.

Release Notes:

- N/A

---------

Co-authored-by: Mikayla <mikayla@zed.dev>
Co-authored-by: Conrad <conrad@zed.dev>

crates/assistant/src/assistant.rs

@@ -209,7 +209,7 @@ pub fn init(
     })
     .detach();
 
-    context_store::init(&client);
+    context_store::init(&client.clone().into());
     prompt_library::init(cx);
     init_language_model_settings(cx);
     assistant_slash_command::init(cx);