[identity] [10/n] Replace usages of Arc<Mutex<SplitSink<WebSocketStream<Upgraded>, Message>>> with custom type WebsocketSink

Summary:
Utilizing a custom type, replaces Arc<Mutex<SplitSink<WebSocketStream<Upgraded>.

Depends on D10878

Test Plan: built with cargo

Reviewers: varun, bartek

Reviewed By: varun

Subscribers: ashoat, tomek

Differential Revision: https://phab.comm.dev/D10903