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/llm/chat.rs

@@ -14,6 +14,16 @@ pub struct ProviderConfig {
     pub model: String,
     pub base_url: Option<String>,
     pub enable_tools: Option<bool>,
+    /// Claude Code session ID for conversation resumption.
+    pub session_id: Option<String>,
+}
+
+/// Result of a chat call, including messages and optional metadata.
+#[allow(dead_code)]
+pub struct ChatResult {
+    pub messages: Vec<Message>,
+    /// Session ID returned by Claude Code for resumption.
+    pub session_id: Option<String>,
 }
 
 fn get_anthropic_api_key_exists_impl(store: &dyn StoreOps) -> bool {
@@ -172,7 +182,7 @@ pub async fn chat<F, U>(
     store: &dyn StoreOps,
     mut on_update: F,
     mut on_token: U,
-) -> Result<Vec<Message>, String>
+) -> Result<ChatResult, String>
 where
     F: FnMut(&[Message]) + Send,
     U: FnMut(&str) + Send,
@@ -219,6 +229,7 @@ where
             .chat_stream(
                 &user_message,
                 &project_root.to_string_lossy(),
+                config.session_id.as_deref(),
                 &mut cancel_rx,
                 |token| on_token(token),
             )
@@ -235,7 +246,10 @@ where
         let mut result = messages.clone();
         result.push(assistant_msg);
         on_update(&result);
-        return Ok(result);
+        return Ok(ChatResult {
+            messages: result,
+            session_id: response.session_id,
+        });
     }
 
     let tool_defs = get_tool_definitions();
@@ -353,7 +367,10 @@ where
         }
     }
 
-    Ok(new_messages)
+    Ok(ChatResult {
+        messages: new_messages,
+        session_id: None,
+    })
 }
 
 async fn execute_tool(call: &ToolCall, state: &SessionState) -> String {