fix(pageserver): categorize client error in basebackup metrics (#9110)

We separated client error from basebackup error log lines in
https://github.com/neondatabase/neon/pull/7523, but we didn't do
anything for the metrics. In this patch, we fixed it.

ref https://github.com/neondatabase/neon/issues/8970

## Summary of changes

We use the same criteria as in `log_query_error` producing an info line
(instead of error) for the metrics. We added a `client_error` category
for the basebackup query time metrics.

---------

Signed-off-by: Alex Chi Z <chi@neon.tech>

libs/postgres_backend/src/lib.rs

@@ -984,6 +984,7 @@ pub fn short_error(e: &QueryError) -> String {
 }
 
 fn log_query_error(query: &str, e: &QueryError) {
+    // If you want to change the log level of a specific error, also re-categorize it in `BasebackupQueryTimeOngoingRecording`.
     match e {
         QueryError::Disconnected(ConnectionError::Io(io_error)) => {
             if is_expected_io_error(io_error) {