Write an item to the server-side LangGraph Store.
Notes:
A failed write is logged at debug and re-raised. The re-raise is
load-bearing: callers (awrite_approval_mode and its callers)
depend on the failure propagating so they can fail closed — drop
the live approval-mode key and interrupt rather than keep
auto-approving. Removing the raise would turn the debug log into
a silent-failure hole, so the higher-severity logging is left to
those callers, which re-log at warning with exc_info.
JSON-serializable item value.
