chore: [release-2.9.x] ci: Update workflows to use actions that don't need organization secrets

[loki-gh-app]

Backport e899826 from #13991

---

What this PR does / why we need it:

Each repository can only have 100 organization secrets and there are now more than 100 in our organization which causes inconsistent behavior.

Some repositories don't have the secrets they need assigned.

These composite actions use secrets stored in Vault that are available to all repositories.

• publish-technical-documentation-next.yml has been tested with https://github.com/grafana/writers-toolkit/blob/main/.github/workflows/publish-technical-documentation.yml.
• publish-technical-documentation-release.yml has been tested with https://github.com/grafana/backend-enterprise/blob/gem-release-2.13/.github/workflows/publish-technical-documentation-release.yml.

Which issue(s) this PR fixes:
Fixes https://github.com/grafana/technical-documentation/issues/979

Special notes for your reviewer:

There is some copy-paste involved in the creation of these workflows. Please check:

For publish-technical-documentation-next.yml:

• The on.push branches and paths filters are correct for your repository.
• The jobs.sync.if repository matches your repository.
• The jobs.sync.steps[1].with.website_directory matches the directory you publish to in the website repository.

For publish-technical-documentation-release.yml:

• The on.push branches, tags, and paths filters are correct for your repository.
• The jobs.sync.if repository matches your repository.
• The jobs.sync.steps[1].with.release_tag_regexp regular expression matches your tags and captures major, minor, and patch versions from those tags.
• The jobs.sync.steps[1].with.release_branch_regexp regular expression matches your release branch names and captures major and minor versions from those branch names.
• The jobs.sync.steps[1].with.release_branch_with_patch_regexp regular expression matches your release branch names if they were to include a patch version, and that it would capture major, minor, and patch versions from those branch names.
• The jobs.sync.steps[1].with.website_directory matches the directory you publish to in the website repository.

We'll also need to backport this to any branches where you are maintaining documentation that you want synced to the website.