Yes it does appear to send that information from the Server to the Client, (parameter "M":"clientMethod"
i.e. "M":"SetChatModeType"
). This article dissects how the protocol works.
Example message format, explained in the article.
Client –> Server
{"H":"hubName", "M":"MethodName", "A":[arguments], "I":messageNumber }
Server –> Client
{"C":"B,6|O,0|P,0|Q,0", "M":[{"H":"hubName", "M":"clientMethod", "A":[arguments]}]}
The relevant response DTO, that triggers the client method is defined here.
As @PatrickFletcher points out, this article is a good resource for increasing SignalR performance and specifically addresses the issue of message size and how to reduce it.