chore: remove unused mode parameter from session-alive handler

The mode parameter is not used by the server, only tracked client-side.

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Co-Authored-By: Claude <noreply@anthropic.com>

sources/app/api.ts

@@ -1166,7 +1166,6 @@ export async function startApi(): Promise<{ app: FastifyInstance; io: Server }>
             sid: string;
             time: number;
             thinking?: boolean;
-            mode?: 'local' | 'remote';
         }) => {
             try {
                 // Basic validation