[proxy] Classify "database not found" errors as user errors (#12603)

## Problem

If a user provides a wrong database name in the connection string, it
should be logged as a user error, not postgres error.

I found 4 different places where we log such errors:
1. `proxy/src/stream.rs:193`, e.g.:
```
{"timestamp":"2025-07-15T11:33:35.660026Z","level":"INFO","message":"forwarding error to user","fields":{"kind":"postgres","msg":"database \"[redacted]\" does not exist"},"spans":{"connect_request#9":{"protocol":"tcp","session_id":"ce1f2c90-dfb5-44f7-b9e9-8b8535e8b9b8","conn_info":"[redacted]","ep":"[redacted]","role":"[redacted]"}},"thread_id":22,"task_id":"370407867","target":"proxy::stream","src":"proxy/src/stream.rs:193","extract":{"ep":"[redacted]","session_id":"ce1f2c90-dfb5-44f7-b9e9-8b8535e8b9b8"}}
```
2. `proxy/src/pglb/mod.rs:137`, e.g.:
```
{"timestamp":"2025-07-15T11:37:44.340497Z","level":"WARN","message":"per-client task finished with an error: Couldn't connect to compute node: db error: FATAL: database \"[redacted]\" does not exist","spans":{"connect_request#8":{"protocol":"tcp","session_id":"763baaac-d039-4f4d-9446-c149e32660eb","conn_info":"[redacted]","ep":"[redacted]","role":"[redacted]"}},"thread_id":14,"task_id":"866658139","target":"proxy::pglb","src":"proxy/src/pglb/mod.rs:137","extract":{"ep":"[redacted]","session_id":"763baaac-d039-4f4d-9446-c149e32660eb"}}
```
3. `proxy/src/serverless/mod.rs:451`, e.g. (note that the error is
repeated 4 times — retries?):
```
{"timestamp":"2025-07-15T11:37:54.515891Z","level":"WARN","message":"error in websocket connection: Couldn't connect to compute node: db error: FATAL: database \"[redacted]\" does not exist: Couldn't connect to compute node: db error: FATAL: database \"[redacted]\" does not exist: db error: FATAL: database \"[redacted]\" does not exist: FATAL: database \"[redacted]\" does not exist","spans":{"http_conn#8":{"conn_id":"ec7780db-a145-4f0e-90df-0ba35f41b828"},"connect_request#9":{"protocol":"ws","session_id":"1eaaeeec-b671-4153-b1f4-247839e4b1c7","conn_info":"[redacted]","ep":"[redacted]","role":"[redacted]"}},"thread_id":10,"task_id":"366331699","target":"proxy::serverless","src":"proxy/src/serverless/mod.rs:451","extract":{"conn_id":"ec7780db-a145-4f0e-90df-0ba35f41b828","ep":"[redacted]","session_id":"1eaaeeec-b671-4153-b1f4-247839e4b1c7"}}
```
4. `proxy/src/serverless/sql_over_http.rs:219`, e.g.
```
{"timestamp":"2025-07-15T10:32:34.866603Z","level":"INFO","message":"forwarding error to user","fields":{"kind":"postgres","error":"could not connect to postgres in compute","msg":"database \"[redacted]\" does not exist"},"spans":{"http_conn#19":{"conn_id":"7da08203-5dab-45e8-809f-503c9019ec6b"},"connect_request#5":{"protocol":"http","session_id":"68387f1c-cbc8-45b3-a7db-8bb1c55ca809","conn_info":"[redacted]","ep":"[redacted]","role":"[redacted]"}},"thread_id":17,"task_id":"16432250","target":"proxy::serverless::sql_over_http","src":"proxy/src/serverless/sql_over_http.rs:219","extract":{"conn_id":"7da08203-5dab-45e8-809f-503c9019ec6b","ep":"[redacted]","session_id":"68387f1c-cbc8-45b3-a7db-8bb1c55ca809"}}
```

This PR directly addresses 1 and 4. I _think_ it _should_ also help with
2 and 3, although in those places we don't seem to log `kind`, so I'm
not quite sure. I'm also confused why in 3 the error is repeated
multiple times.

## Summary of changes

Resolves https://github.com/neondatabase/neon/issues/9440

libs/proxy/tokio-postgres2/src/error/mod.rs

@@ -9,7 +9,7 @@ use postgres_protocol2::message::backend::{ErrorFields, ErrorResponseBody};
 pub use self::sqlstate::*;
 
 #[allow(clippy::unreadable_literal)]
-mod sqlstate;
+pub mod sqlstate;
 
 /// The severity of a Postgres error or notice.
 #[derive(Debug, Copy, Clone, PartialEq, Eq)]

proxy/src/compute/mod.rs

@@ -8,6 +8,7 @@ use futures::{FutureExt, TryFutureExt};
 use itertools::Itertools;
 use postgres_client::config::{AuthKeys, ChannelBinding, SslMode};
 use postgres_client::connect_raw::StartupStream;
+use postgres_client::error::SqlState;
 use postgres_client::maybe_tls_stream::MaybeTlsStream;
 use postgres_client::tls::MakeTlsConnect;
 use thiserror::Error;
@@ -22,7 +23,7 @@ use crate::context::RequestContext;
 use crate::control_plane::client::ApiLockError;
 use crate::control_plane::errors::WakeComputeError;
 use crate::control_plane::messages::MetricsAuxInfo;
-use crate::error::{ReportableError, UserFacingError};
+use crate::error::{ErrorKind, ReportableError, UserFacingError};
 use crate::metrics::{Metrics, NumDbConnectionsGuard};
 use crate::pqproto::StartupMessageParams;
 use crate::proxy::connect_compute::TlsNegotiation;
@@ -65,12 +66,13 @@ impl UserFacingError for PostgresError {
 }
 
 impl ReportableError for PostgresError {
-    fn get_error_kind(&self) -> crate::error::ErrorKind {
+    fn get_error_kind(&self) -> ErrorKind {
         match self {
-            PostgresError::Postgres(e) if e.as_db_error().is_some() => {
-                crate::error::ErrorKind::Postgres
-            }
-            PostgresError::Postgres(_) => crate::error::ErrorKind::Compute,
+            PostgresError::Postgres(err) => match err.as_db_error() {
+                Some(err) if err.code() == &SqlState::INVALID_CATALOG_NAME => ErrorKind::User,
+                Some(_) => ErrorKind::Postgres,
+                None => ErrorKind::Compute,
+            },
         }
     }
 }
@@ -110,9 +112,9 @@ impl UserFacingError for ConnectionError {
 }
 
 impl ReportableError for ConnectionError {
-    fn get_error_kind(&self) -> crate::error::ErrorKind {
+    fn get_error_kind(&self) -> ErrorKind {
         match self {
-            ConnectionError::TlsError(_) => crate::error::ErrorKind::Compute,
+            ConnectionError::TlsError(_) => ErrorKind::Compute,
             ConnectionError::WakeComputeError(e) => e.get_error_kind(),
             ConnectionError::TooManyConnectionAttempts(e) => e.get_error_kind(),
             #[cfg(test)]

proxy/src/serverless/backend.rs

@@ -4,6 +4,7 @@ use std::time::Duration;
 use ed25519_dalek::SigningKey;
 use hyper_util::rt::{TokioExecutor, TokioIo, TokioTimer};
 use jose_jwk::jose_b64;
+use postgres_client::error::SqlState;
 use postgres_client::maybe_tls_stream::MaybeTlsStream;
 use rand_core::OsRng;
 use tracing::field::display;
@@ -459,15 +460,14 @@ impl ReportableError for HttpConnError {
         match self {
             HttpConnError::ConnectError(_) => ErrorKind::Compute,
             HttpConnError::ConnectionClosedAbruptly(_) => ErrorKind::Compute,
-            HttpConnError::PostgresConnectionError(p) => {
-                if p.as_db_error().is_some() {
-                    // postgres rejected the connection
-                    ErrorKind::Postgres
-                } else {
-                    // couldn't even reach postgres
-                    ErrorKind::Compute
-                }
-            }
+            HttpConnError::PostgresConnectionError(p) => match p.as_db_error() {
+                // user provided a wrong database name
+                Some(err) if err.code() == &SqlState::INVALID_CATALOG_NAME => ErrorKind::User,
+                // postgres rejected the connection
+                Some(_) => ErrorKind::Postgres,
+                // couldn't even reach postgres
+                None => ErrorKind::Compute,
+            },
             HttpConnError::LocalProxyConnectionError(_) => ErrorKind::Compute,
             HttpConnError::ComputeCtl(_) => ErrorKind::Service,
             HttpConnError::JwtPayloadError(_) => ErrorKind::User,

proxy/src/serverless/sql_over_http.rs

@@ -192,34 +192,29 @@ pub(crate) async fn handle(
             let line = get(db_error, |db| db.line().map(|l| l.to_string()));
             let routine = get(db_error, |db| db.routine());
 
-            match &e {
-                SqlOverHttpError::Postgres(e)
-                    if e.as_db_error().is_some() && error_kind == ErrorKind::User =>
-                {
-                    // this error contains too much info, and it's not an error we care about.
-                    if tracing::enabled!(Level::DEBUG) {
-                        tracing::debug!(
-                            kind=error_kind.to_metric_label(),
-                            error=%e,
-                            msg=message,
-                            "forwarding error to user"
-                        );
-                    } else {
-                        tracing::info!(
-                            kind = error_kind.to_metric_label(),
-                            error = "bad query",
-                            "forwarding error to user"
-                        );
-                    }
-                }
-                _ => {
-                    tracing::info!(
+            if db_error.is_some() && error_kind == ErrorKind::User {
+                // this error contains too much info, and it's not an error we care about.
+                if tracing::enabled!(Level::DEBUG) {
+                    debug!(
                         kind=error_kind.to_metric_label(),
                         error=%e,
                         msg=message,
                         "forwarding error to user"
                     );
+                } else {
+                    info!(
+                        kind = error_kind.to_metric_label(),
+                        error = "bad query",
+                        "forwarding error to user"
+                    );
                 }
+            } else {
+                info!(
+                    kind=error_kind.to_metric_label(),
+                    error=%e,
+                    msg=message,
+                    "forwarding error to user"
+                );
             }
 
             json_response(