Duo Chat Agent Selection Issues

Three related issues affecting the agent selection experience in Duo Chat:

Issue 1: Panel only opens when non-active agent is selected

Current behavior:

• When Duo Chat panel is closed/hidden
• Selecting a non-default agent (e.g., "Pirate") opens the panel ✓
• Selecting the default agent ("GitLab Duo") does NOT open the panel

Expected behavior:

• Selecting ANY agent from the dropdown should open the panel if it's currently closed

Demo: https://gitlab.slack.com/archives/C085M5071LG/p1765336286045459

Issue 2: Default agent displays inconsistent name after reload

Current behavior:

1. Select "GitLab Duo" from agent dropdown → displays "GitLab Duo" ✓
2. Reload the page → displays "Duo Agent" ✗

Expected behavior:

• Should consistently display "GitLab Duo" both before and after reload

Issue 3: Cannot reselect the same agent from dropdown

Current behavior:

1. Select "Planner" agent → works ✓
2. Click "New Chat" button
3. Try to select "Planner" again → doesn't work ✗
4. Must select a different agent first, then can select "Planner"

Expected behavior:

• Should be able to select the same agent multiple times in a row for new chats