fix(web): persist thinking, title sessions distinctly, and send follow-ups from the answer

Thinking disappeared on failure and never reached session storage. Keep the
sanitized chain on disk and on errors, and regroup the sidebar around reports,
charts, favorites, and dated history titles.

Co-authored-by: Cursor <cursoragent@cursor.com>

tests/test_session_management_entrypoints.py

@@ -23,13 +23,11 @@ def test_chat_history_management_actions_are_exposed() -> None:
         "shareSession",
         "share_payload_version",
         "messages.map",
-        "onContextMenu",
-        "session-menu-trigger",
-        'role="menu"',
-        "closeSessionMenu",
-        "Escape",
-        "focus-within",
-        "置顶",
+            "onContextMenu",
+            "session-menu-trigger",
+            "Escape",
+            "focus-within",
+            "收藏",
         "重命名",
         "归档",
         "恢复",