Accept story 39: Persistent Claude Code Sessions in Web UI

Use --resume <session_id> with claude -p so the web UI claude-code-pty
provider maintains full conversation context across messages, identical
to a long-running terminal Claude Code session.

Changes:
- Capture session_id from claude -p stream-json system event
- Pass --resume on subsequent messages in same chat session
- Thread session_id through ProviderConfig, ChatResult, WsResponse
- Frontend stores sessionId per chat, clears on New Session
- Unset CLAUDECODE env to allow nested spawning from server
- Wait for clean process exit to ensure transcript flush to disk

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

server/src/http/ws.rs

@@ -33,6 +33,8 @@ enum WsRequest {
 enum WsResponse {
     Token { content: String },
     Update { messages: Vec<Message> },
+    /// Session ID for Claude Code conversation resumption.
+    SessionId { session_id: String },
     Error { message: String },
 }
 
@@ -83,8 +85,15 @@ pub async fn ws_handler(ws: WebSocket, ctx: Data<&Arc<AppContext>>) -> impl poem
                         )
                         .await;
 
-                        if let Err(err) = result {
-                            let _ = tx.send(WsResponse::Error { message: err });
+                        match result {
+                            Ok(chat_result) => {
+                                if let Some(sid) = chat_result.session_id {
+                                    let _ = tx.send(WsResponse::SessionId { session_id: sid });
+                                }
+                            }
+                            Err(err) => {
+                                let _ = tx.send(WsResponse::Error { message: err });
+                            }
                         }
                     }
                     Ok(WsRequest::Cancel) => {