why

Applications that centralize Browserbase setup need to reuse an existing
@browserbasehq/sdk client when launching or attaching Stagehand browsers.

Today, browserbase.launch() and browserbase.connect() always create a new
Browserbase client. This prevents callers from supplying their own configured
Browserbase management client and endpoint.

what changed

• Add an optional client?: Browserbase option to TypeScript
  browserbase.launch() and browserbase.connect().
• Reuse the supplied client for Browserbase session and extension lifecycle
  operations, including session create, retrieve, release, and extension
  provisioning/cleanup.
• Preserve existing behavior when client is omitted.
• Keep apiKey for Stagehand runtime authentication.
• Document that the supplied client controls Browserbase management requests;
  baseUrl is therefore not used for those requests.
• Add TypeScript factory, lifecycle, contract, package-consumer, and
  cross-language schema-parity coverage.
• Add a minor Changeset for @browserbasehq/stagehand.

test plan

• pnpm build
• pnpm fmt:check
• pnpm check
• pnpm test — 33 workspace tasks passed; 80 test files / 627 tests
• Focused SDK coverage — 20 test files / 282 tests
• Published-package consumer type fixture
• Cross-language schema-parity test
• Go SDK test, vet, build, generator, and example checks
• Full just Python coverage was not run because the disposable validation
  sandbox did not include uv.
• The Go extension-pack consistency check reports a pre-existing mismatch
  in the committed embedded extension archive; this PR changes neither
  artifact.

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Summary by cubic

Adds an optional client option to browserbase.launch() and browserbase.connect() so callers can reuse an existing @browserbasehq/sdk client for session and extension lifecycle management. Previously these methods always created a new Browserbase client; now they use the supplied client, and existing behavior is unchanged when client is omitted.

• When client is provided, it determines the Browserbase management endpoint; baseUrl is ignored for those calls.
• apiKey still configures Stagehand runtime authentication.