proxy small tweaks (#6398)

## Problem

In https://github.com/neondatabase/neon/pull/6283 I did a couple changes
that weren't directly related to the goal of extracting the state
machine, so I'm putting them here

## Summary of changes

- move postgres vs console provider into another enum
- reduce error cases for link auth
- slightly refactor link flow

proxy/src/console/mgmt.rs

@@ -13,16 +13,10 @@ use tracing::{error, info, info_span, Instrument};
 static CPLANE_WAITERS: Lazy<Waiters<ComputeReady>> = Lazy::new(Default::default);
 
 /// Give caller an opportunity to wait for the cloud's reply.
-pub async fn with_waiter<R, T, E>(
+pub fn get_waiter(
     psql_session_id: impl Into<String>,
-    action: impl FnOnce(Waiter<'static, ComputeReady>) -> R,
-) -> Result<T, E>
-where
-    R: std::future::Future<Output = Result<T, E>>,
-    E: From<waiters::RegisterError>,
-{
-    let waiter = CPLANE_WAITERS.register(psql_session_id.into())?;
-    action(waiter).await
+) -> Result<Waiter<'static, ComputeReady>, waiters::RegisterError> {
+    CPLANE_WAITERS.register(psql_session_id.into())
 }
 
 pub fn notify(psql_session_id: &str, msg: ComputeReady) -> Result<(), waiters::NotifyError> {
@@ -77,7 +71,7 @@ async fn handle_connection(socket: TcpStream) -> Result<(), QueryError> {
 }
 
 /// A message received by `mgmt` when a compute node is ready.
-pub type ComputeReady = Result<DatabaseInfo, String>;
+pub type ComputeReady = DatabaseInfo;
 
 // TODO: replace with an http-based protocol.
 struct MgmtHandler;
@@ -102,7 +96,7 @@ fn try_process_query(pgb: &mut PostgresBackendTCP, query: &str) -> Result<(), Qu
     let _enter = span.enter();
     info!("got response: {:?}", resp.result);
 
-    match notify(resp.session_id, Ok(resp.result)) {
+    match notify(resp.session_id, resp.result) {
         Ok(()) => {
             pgb.write_message_noflush(&SINGLE_COL_ROWDESC)?
                 .write_message_noflush(&BeMessage::DataRow(&[Some(b"ok")]))?