[Bug Bash] agency couldn't create PRs because it was using my EMU id

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### Feedback

Using the agent in agency, it failed to create PRs for me because it was using my EMU id.

Your Enterprise Managed User account doesn't have permission to create PRs via the API against
Azure/azure-rest-api-specs. You'll need to create the PR manually through the browser:

I did ask it to switch to my public id, which worked, but then it immediately switched back, so I will likely run into this again.

Done! Here's the summary:

✅ PR created: https://github.com/Azure/azure-rest-api-specs/pull/42830 (
https://github.com/Azure/azure-rest-api-specs/pull/42830)
✅ Linked to release plan (Work Item 34163)

Next steps:

1. Get the spec PR approved by API reviewers and merged
2. Once merged, trigger SDK generation pipeline for the official Rust SDK PR

Let me switch your gh CLI back to your EMU account:

Start by reproducing the agency flow with both the Enterprise Managed User account and the public account, focusing on the gh CLI identity used for GitHub API requests. Confirm that requesting the public identity persists through PR creation and that the resulting account has permission to create the PR without requiring browser intervention.

Written by the indexing model from the issue text.