Update distribution documentation after ownership migrations

## Context

Follow-up to #796 and the distribution ownership migrations:

- Docker: #799
- WinGet: #800
- Homebrew: #801

The public instructions and examples should switch to the organization-owned identities only after each destination exists and has been verified. Transitional compatibility guidance should remain visible where old identifiers cannot redirect automatically.

## Scope

- Update the GitHub Pages installation guide with the canonical Docker, WinGet, and Homebrew identities.
- Update Alpine Linux guidance and Docker examples.
- Correct the README migration table, including the actual unchanged NuGet package ID (`grate`).
- Update local Docker helper output/tags and relevant Docker Compose/Kubernetes base-image examples.
- Update the local WinGet example and installer/submission helper.
- Update the documented release process and credential/ownership boundaries.
- Update the Homebrew tap README/cask metadata and Docker Hub repository descriptions where those changes are not already covered by their migration issues.
- Retain explicit transition instructions for `erikbra/grate`, `erikbra.grate`, and `erikbra/cask/grate` where compatibility is not automatic.
- Check the repository for stale ownership references and classify each remaining occurrence as historical, transitional, or erroneous.

## Acceptance criteria

- [ ] Public installation instructions use only verified, live canonical destinations.
- [ ] Docker, WinGet, and Homebrew commands have been smoke-tested.
- [ ] The README migration table accurately describes GitHub, NuGet, Docker, WinGet, and Homebrew.
- [ ] Examples and local helper scripts match the public documentation.
- [ ] Intentional legacy references are clearly marked as migration/compatibility guidance.
- [ ] Links pass validation and the rendered GitHub Pages site has been checked after publication.

## References

- #796
- #799
- #800
- #801
- Current installation page: https://grate-devs.github.io/grate/getting-grate/