proxy: merge handle_client (#4740)

## Problem

Second half of #4699. we were maintaining 2 implementations of
handle_client.

## Summary of changes

Merge the handle_client code, but abstract some of the details.

## Checklist before requesting a review

- [X] I have performed a self-review of my code.
- [ ] If it is a core feature, I have added thorough tests.
- [ ] Do we need to implement analytics? if so did you add the relevant
metrics to the dashboard?
- [ ] If this PR requires public announcement, mark it with
/release-notes label and add several sentences in this section.

## Checklist before merging

- [ ] Do not forget to reformat commit message to not include the above
checklist

proxy/src/http/websocket.rs

@@ -1,5 +1,8 @@
 use crate::{
-    cancellation::CancelMap, config::ProxyConfig, error::io_error, proxy::handle_ws_client,
+    cancellation::CancelMap,
+    config::ProxyConfig,
+    error::io_error,
+    proxy::{handle_client, ClientMode},
 };
 use bytes::{Buf, Bytes};
 use futures::{Sink, Stream, StreamExt};
@@ -150,12 +153,12 @@ async fn serve_websocket(
     hostname: Option<String>,
 ) -> anyhow::Result<()> {
     let websocket = websocket.await?;
-    handle_ws_client(
+    handle_client(
         config,
         cancel_map,
         session_id,
         WebSocketRw::new(websocket),
-        hostname,
+        ClientMode::Websockets { hostname },
     )
     .await?;
     Ok(())