docs: refresh public tidu rustdoc examples

[shinaoka]

Summary

• Rewrite public tidu rustdoc examples around scalar reverse-mode, forward-mode, HVP, and a custom value type
• Add a regression test that rejects tenferro_ references in public docs
• Wire chainrules explicitly for runnable doctests

Test Plan

• cargo nextest run --release -p tidu --test public_rustdoc_examples
• cargo nextest run --release --workspace --no-fail-fast
• cargo test --doc --release --workspace
• cargo doc --workspace --no-deps
• python3 scripts/check-docs-site.py
• bash scripts/build_docs_site.sh