Fix docs deploy output path and add manual branch dispatch

[Copilot]

The docs deploy workflow currently builds mdBook output outside the repository root, while deploy still publishes out/, which leaves /sdk/out missing at deploy time. This updates mdBook to write into repo-local out/ and adds manual dispatch so maintainers can validate the workflow on a branch before merge.

• Build output path

  • Change the root mdbook script to emit tutorials into out/tutorials under the repository root.
  • Preserve the existing deploy target of FOLDER: out, so the workflow publishes the same top-level docs directory structure without relying on .. path traversal.

• Workflow trigger

  • Add workflow_dispatch to .github/workflows/docs.yml in addition to the existing push trigger on master.
  • This allows maintainers to run the docs deploy workflow manually against a branch for pre-merge verification.

• Resulting behavior

  • mdbook now writes where the deploy step already expects artifacts to exist.

{
  "scripts": {
    "mdbook": "mdbook build tutorials --dest-dir out/tutorials"
  }
}