test: migrate webdriver callers to next.browser (#93941)
### What?
Migrate remaining direct `next-webdriver` test callers that have a
`NextInstance` to `next.browser()`, and expose the shared `Playwright`
browser type from `e2e-utils`.
### Why?
`NextInstance.browser` should be the supported browser-opening interface
for test fixtures, with `next-webdriver` kept as the private
implementation detail.
### How?
Updated affected development, e2e, and production tests to call
`next.browser()` directly, passing `baseUrl` where tests intentionally
target a manually spawned or proxied server. Shared helpers now receive
browser callbacks from the test context, and browser types import
`Playwright` from `e2e-utils` instead of deriving from `next.browser` or
importing from private paths.
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