[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