Update docs-lint job to use latest image.

What does this MR do?

Updates the docs image to make use of the latest docs tools, including a general update to docs site code as well.

Related issues

Related to gitlab#218764 (closed)

Author's checklist (required)

• Follow the Documentation Guidelines and Style Guide.
• If you have developer access or higher (for example, GitLab team members or Core Team members)
  • Apply the documentation label.
  • Assign the designated Technical Writer.

When applicable:

• Update the permissions table.
• Link docs to and from the higher-level index page, plus other related docs where helpful.
• Add GitLab's version history note(s).
• Add/update the feature flag section.
• If you're changing document headings, search docs/* for old headings and replace them with the new ones to avoid broken anchors.

Review checklist

All reviewers can help ensure accuracy, clarity, completeness, and adherence to the Documentation Guidelines and Style Guide.

1. Primary Reviewer

• Review by a code reviewer or other selected colleague to confirm accuracy, clarity, and completeness. This can be skipped for minor fixes without substantive content changes.

2. Technical Writer

• Technical writer review. If not requested for this MR, must be scheduled post-merge. To request for this MR, assign the writer listed for the applicable DevOps stage.
  • Ensure Technical Writing, documentation, and a docs:: scoped label are added.
  • Add docs-only when the only files changed are under docs/*.
  • Add twdoing when starting work on the MR.
  • Add twfinished if Technical Writing team work on the MR is complete but it remains open.

3. Maintainer

1. Review by assigned maintainer, who can always request/require the above reviews. Maintainer's review can occur before or after a technical writer review.
2. Ensure a release milestone is set.
3. If there has not been a technical writer review, create an issue for one using the Doc Review template.