Yes. That protocol is effectively part of HTTP now.
The WebSocket Protocol is an independent TCP-based protocol. Its only relationship to HTTP is that its handshake is interpreted by HTTP servers as an Upgrade request.
If you do connection setup, service disambiguation, redirects, and so on with HTTP and switch to a non-HTTP interactive protocol, what you've effectively done is create an HTTP sub-protocol, and I don't care what the document you quoted says.