dev: include Markdown link checks in the local lint suite

[kumarUjjawal]

Which issue does this PR close?

• Part of Add Local Scripts to Reproduce Full CI #21048. The issue stays open for the other checklist items.

Rationale for this change

dev/rust_lint.sh is the local mirror of the CI lint jobs, but it does not run the Markdown link check. CI runs ci/scripts/markdown_link_check.sh in dev.yml. A developer can push a broken internal link after a clean local lint run and then see the failure only in CI. This PR adds the existing checker to the local suite.

What changes are included in this PR?

• dev/rust_lint.sh runs ci/scripts/markdown_link_check.sh as a read-only step, after the workflow install check and before the Rust documentation build. The --write and --allow-dirty flags never reach it.
• The runner loads LYCHEE_VERSION from ci/scripts/utils/tool_versions.sh and installs that version when lychee is missing. An installed lychee is used as is, which is the runner's existing policy for other tools.
• ci/scripts/markdown_link_check.sh becomes executable, because the runner invokes each registered script directly. The script body, lychee.toml, the file selection, and the GitHub workflow do not change.
• docs/source/contributor-guide/testing.md documents the new behavior and keeps the standalone instructions.

What is the testing strategy for this PR?

• A disposable fixture with stubbed steps and tools: the checker runs once with no arguments in check mode and in both write modes, a missing lychee triggers exactly the pinned install command, a present lychee triggers no install, and a checker failure stops the suite before the later steps.
• A disposable clone with an injected broken internal link: the standalone checker and ./dev/rust_lint.sh both fail with the same exit code and name the file. After the fix, the full suite passes.
• The real checker and the full ./dev/rust_lint.sh pass on this branch with lychee 0.23.0.

Are there any user-facing changes?

No. Developers get the link check in the local lint suite. CI is unchanged.

🤖 Generated with Claude Code

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 81.90%. Comparing base (c2cf289) to head (0c1ac9d).

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #25287      +/-   ##
==========================================
- Coverage   81.90%   81.90%   -0.01%     
==========================================
  Files        1134     1134              
  Lines      425261   425261              
  Branches   425261   425261              
==========================================
- Hits       348325   348320       -5     
- Misses      56295    56297       +2     
- Partials    20641    20644       +3     

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[kumarUjjawal]

Thank you @martin-g for the review.