[Task] Send copilot-sdk as the default SDK integration ID

## Outcome

The Copilot SDK launches its bundled runtime with `Copilot-Integration-Id: copilot-sdk` for default SDK traffic while preserving explicit registered integration overrides.

## Current problem

The SDK currently inherits the direct CLI default and sends `copilot-developer-cli` unless the host sets `GITHUB_COPILOT_INTEGRATION_ID`. CAPI therefore classifies default SDK traffic as direct CLI traffic. `Copilot-Harness-Id: copilot-sdk` does not solve this because the SDK and standalone CLI share that harness value.

## SDK team ownership

The Copilot SDK team owns the launch-path behavior in this issue. Assign an SDK engineering DRI before implementation and link the implementation pull requests for every affected language or shared launcher.

## Scope

- Set `GITHUB_COPILOT_INTEGRATION_ID=copilot-sdk` when the SDK launches the bundled CLI or runtime and the host did not provide an explicit integration override
- Preserve host-provided registered integration IDs exactly
- Apply the behavior consistently across supported SDK language surfaces or in the shared launch layer used by them
- Keep standalone Copilot CLI defaulting to `copilot-developer-cli`
- Keep `Copilot-Harness-Id` as supporting telemetry only
- Coordinate rollout and compatibility normalization with github/copilot-api#37988

## Acceptance criteria

- [ ] Default SDK requests send `Copilot-Integration-Id: copilot-sdk`
- [ ] A host-provided registered integration override remains effective
- [ ] Standalone CLI requests continue to send `copilot-developer-cli`
- [ ] Tests cover default behavior and explicit overrides across supported SDK launch paths
- [ ] An SDK engineering DRI and implementation pull request are linked here
- [ ] CAPI telemetry confirms default SDK traffic transitions from `copilot-developer-cli` to `copilot-sdk`

## Dependencies and references

- Parent epic github/copilot#30409
- SDK App-token plumbing github/copilot#30423
- CAPI integration allowlist and migration github/copilot-api#37988
- ADR github/copilot#30449

Start by locating the bundled CLI/runtime launch path and the tests for integration headers, then compare default and host-provided override handling across supported SDK surfaces. Done means default SDK traffic uses the SDK integration ID, explicit overrides remain unchanged, standalone CLI behavior is preserved, and the required tests, DRI, implementation links, and telemetry confirmation are present.

索引モデルが issue の本文から書いたものです。