_get_backend keys its cache on permission mode only, so a cua-driver
cached before the Bot Desktop started (or restarted on another display
number) keeps acting on the old seat, or on no display. desktop_identity()
returns the DISPLAY a fresh spawn would get and backend_display_stale()
compares it with the identity recorded at cache time; tool.py (owned by
another lane) records the identity next to _backend_permission_modes[sid]
and detaches+stops a stale backend the way the /yolo mode swap does.
(cherry picked from commit 896304bcbb98b12f821c6e19f10b9f174fa08889)
86cf1d760f