fix(ai-help): show stopped message once + restore search animation

[caugner]

Summary

Problem

Recent changes caused two regressions:

1. The "Stopped answering" message was potentially showing multiple times.
2. The icon next to "Searching for MDN content…" was no longer blinking.

Solution

1. Move the "Stopped " message out of CSS into React.
2. Add the missing active class for the search status.

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Screenshots

Before

After

Screen.Recording.2024-03-01.at.15.29.15.mov

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How did you test this change?

Asked a question locally and stopped when it was generating content in a nested list.