pageserver: shut down compute connections at libpq level (#12642)

## Problem

Previously, if a get page failure was cause by timeline shutdown, the
pageserver would attempt to tear down the connection gracefully:
`shutdown(SHUT_WR)` followed by `close()`.

This triggers a code path on the compute where it has to tell apart
between an idle connection and a closed one. That code is bug prone, so
we can just side-step the issue by shutting down the connection via a
libpq error message.

This surfaced as instability in test_shard_resolve_during_split_abort.
It's a new test, but the issue existed for ages.

## Summary of Changes

Send a libpq error message instead of doing graceful TCP connection
shutdown.

Closes LKB-648

libs/postgres_backend/src/lib.rs

@@ -749,7 +749,18 @@ impl<IO: AsyncRead + AsyncWrite + Unpin> PostgresBackend<IO> {
                 trace!("got query {query_string:?}");
                 if let Err(e) = handler.process_query(self, query_string).await {
                     match e {
-                        QueryError::Shutdown => return Ok(ProcessMsgResult::Break),
+                        err @ QueryError::Shutdown => {
+                            // Notify postgres of the connection shutdown at the libpq
+                            // protocol level. This avoids postgres having to tell apart
+                            // from an idle connection and a stale one, which is bug prone.
+                            let shutdown_error = short_error(&err);
+                            self.write_message_noflush(&BeMessage::ErrorResponse(
+                                &shutdown_error,
+                                Some(err.pg_error_code()),
+                            ))?;
+
+                            return Ok(ProcessMsgResult::Break);
+                        }
                         QueryError::SimulatedConnectionError => {
                             return Err(QueryError::SimulatedConnectionError);
                         }