fix: send correct connection scopes for client (#20312)

## Description

When a client joins with AzureClient, it will always try to join in
`write` even if it only has `read` permissions according to its token
claims. On the server (Nexus Lambda), we handle that correctly by
creating a read-scoped connection. However, the server would incorrectly
handle client details and send connection info like this:

```js
socket.emit("connect_document_success", { /* ... */, mode: "read");
// ...
socket.emitToRoom(
    "tenantId/documentId",
    "signal",
    { 
       /* ... join signal message ...*/,
       client: { /* ... */, mode: "write"
    });
```

I'm not sure about the specifics of what's happening on the client, but
when testing read client Audience and Signals usage in #20042, I
realized that the read client audience is not working appropriately
without this change. My understanding is it's because Audience is trying
to use Quorum (Join Ops) instead of Join Signals because the Join Signal
client details said `{ mode: "write" }` even though Connection details
said `{ mode: "read" }`

server/routerlicious/packages/lambdas/src/nexus/index.ts

@@ -427,6 +427,7 @@ export function configureWebSocketServices(
 			const isSummarizer = messageClient.details?.type === summarizerClientType;
 			messageClient.user = claims.user;
 			messageClient.scopes = claims.scopes;
+			messageClient.mode = isWriter(claims.scopes, message.mode) ? "write" : "read";
 
 			// 1. Do not give SummaryWrite scope to clients that are not summarizers.
 			// 2. Store connection timestamp for all clients but the summarizer.