Split public and browser WS server

[sheremet-va]

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

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• Read the Contributing Guidelines.
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[sheremet-va]

@userquin I would recommend doing this. We have a workaround with browserPromise, but in fact, we can just use a regular RPC. Here is the line that setups a WebSocket:

vitest/packages/vitest/src/integrations/browser/server.ts

Line 63 in db22c67

;(await import('../../api/setup')).setup(project, server)

If we create a separate RPC, we can share it with the browser provider and reuse it in the pool to wait for a response.