Improve AI agents live PTY prompt hardening

## Summary

Follow-up from #9102 review: improve how the Azure AI Agents live E2E tests handle PTY prompts and external-tool prompt edge cases.

## Context

#9102 fixed the known live failure modes where stale PTY screen content was treated as an active prompt, and where GitHub auth/token prompts could cause the driver to continue interacting incorrectly. The current fix is targeted and validated, but it still relies on string-based PTY detection.

## Follow-up ideas

1. If the init prompt flow changes, cover the prompt sequence and selection logic with fake-console/unit/integration tests that share the same sequence/prompts used by the live access pipeline.
2. Add fast PTY-driver tests for terminal/GitHub/external-tool prompt edge cases without provisioning Azure resources, so unexpected cases can be exposed cheaply and repeatedly.
3. Keep live access focused on the Azure golden path, and promote validated PTY-driver hardening into live only after it is covered by faster tests.

Start by reading the fixes and review context in #9102, then trace the Azure AI Agents live E2E access pipeline and its PTY driver. Add fake-console or unit/integration coverage for init prompt sequencing and fast tests for terminal, GitHub, and external-tool prompt edge cases. Done means the Azure live path stays focused on its golden path while validated PTY hardening is covered by faster tests.

Written by the indexing model from the issue text.