Clear and concise description of the problem

Currently, browser and public-facing API share the same web socket reporter. We should limit public usage and move browser methods into their own reporter so it's easier to manage and not break public API.

Suggested solution

Move browser methods from packages/vitest/src/api/setup.ts into their own reporter (or better just as create a separate rpc).

Alternative

No response

Additional context

No response

Validations

• Follow our Code of Conduct
• Read the Contributing Guidelines.
• Read the docs.
• Check that there isn't already an issue that request the same feature to avoid creating a duplicate.