fix(chat): include user object in POST /api/chat response

- Changed create action to render show.rabl view instead of raw model
- Response now includes nested user object with name field
- Matches format returned by GET /api/chat (index action)
- Fixes 'Unknown' sender name issue for sent messages

Follows lobby chat pattern for consistent API response format.

web/app/controllers/api_chats_controller.rb

@@ -8,7 +8,12 @@ class ApiChatsController < ApiController
 
     @chat_msg = ChatMessage.create(current_user, @music_session, params[:message], params[:channel], params[:client_id], @target_user, @lesson_session)
 
-    respond_with_model(@chat_msg)
+    if @chat_msg.errors.any?
+      respond_with @chat_msg, status: :unprocessable_entity, layout: nil
+    else
+      @chat = @chat_msg
+      render "api_chats/show", :layout => nil, :status => 200
+    end
   end
 
   def index