[Studio] fix: connect user logout to API

[Loyal-Young]

Summary

• wire the account dropdown's logout action to the existing backend logout endpoint
• clear the local authentication state even when the backend is unreachable, preventing a stale client session
• make the profile menu entry open the settings page and add auth request-contract coverage

Validation

• npm.cmd test -- auth.test.ts
• The local lint and TypeScript commands were retried separately but timed out without diagnostics on this Windows host; no Node child process remained afterward.