fix(clippy): Store exit code to respect errors

[blorente]

Assume this change in rules_rust is going to merge, and take advantage of it to keep the clippy exit code in a file and avoid differentiating between warnings and errors.

Changes are visible to end-users: yes

• Searched for relevant documentation and updated as needed: yes
• Breaking change (forces users to change their own code or config): no
• Suggested release notes appear below: no (clippy is not released yet, so we don't include notes yet)

Test plan

• New test cases added (but manual)

[aspect-workflows]

Bazel 7 (Test)

All tests were cache hits

4 tests (100.0%) were fully cached saving 2s.

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Bazel 8 (Test)

All tests were cache hits

4 tests (100.0%) were fully cached saving 2s.

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Bazel 9 (Test)

All tests were cache hits

4 tests (100.0%) were fully cached saving 1s.