Move Browser Playwright cache to tmp

Use /a0/tmp/playwright as the Browser plugin Chromium cache and Docker install target while preserving full Chromium installs.

Add startup migration cleanup for retired usr Playwright caches, update Browser status/runtime references and docs, and cover migration behavior with focused regressions.

plugins/_browser/extensions/python/startup_migration/_20_browser_playwright_cache.py

@@ -0,0 +1,45 @@
+from __future__ import annotations
+
+import threading
+from typing import Any
+
+from helpers.extension import Extension
+from helpers.print_style import PrintStyle
+from plugins._browser import hooks
+
+
+_startup_migration_thread: threading.Thread | None = None
+
+
+class BrowserPlaywrightCacheMigration(Extension):
+    def execute(self, **kwargs):
+        _start_background_cache_migration()
+
+
+def _start_background_cache_migration() -> threading.Thread:
+    global _startup_migration_thread
+
+    if _startup_migration_thread and _startup_migration_thread.is_alive():
+        return _startup_migration_thread
+
+    _startup_migration_thread = threading.Thread(
+        target=_migrate_cache_safely,
+        name="a0-browser-playwright-cache-migration",
+        daemon=True,
+    )
+    _startup_migration_thread.start()
+    return _startup_migration_thread
+
+
+def _migrate_cache_safely() -> None:
+    try:
+        _log_cache_migration_result(hooks.cleanup_playwright_cache())
+    except Exception as exc:
+        PrintStyle.warning("Browser Playwright cache migration failed:", exc)
+
+
+def _log_cache_migration_result(result: dict[str, Any]) -> None:
+    if result.get("errors"):
+        PrintStyle.warning("Browser Playwright cache migration reported errors:", result["errors"])
+    elif result.get("migrated") or result.get("removed"):
+        PrintStyle.info("Browser Playwright cache prepared:", result)