ref(openai): Separate input handling

[alexander-alderman-webb]

Description

For #5359, it is easier to have distinct handling of the messages and input types for completions and responses APIs, respectively.

Issues

Reminders

• Please add tests to validate your changes, and lint your code using tox -e linters.
• Add GH Issue ID & Linear ID (if applicable)
• PR title should use conventional commit style (feat:, fix:, ref:, meta:)
• For external contributors: CONTRIBUTING.md, Sentry SDK development docs, Discord community

[github-actions]

Semver Impact of This PR

🟢 Patch (bug fixes)

📋 Changelog Preview

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Entries from this PR are highlighted with a left border (blockquote style).

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New Features ✨

• feat(anthropic): Set system instruction attribute by alexander-alderman-webb in #5353
• feat(google-genai): Set system instruction attribute by alexander-alderman-webb in #5354
• feat(langchain): Set system instruction attribute by alexander-alderman-webb in #5357
• feat(openai-agents): Set system instruction attribute by alexander-alderman-webb in #5355
• feat(pydantic-ai): Set system instruction attribute by alexander-alderman-webb in #5356

Internal Changes 🔧

• ref(openai): Separate input handling by alexander-alderman-webb in #5358

• ci(release): Fix changelog-preview permissions by BYK in #5368
• ci: Fix path in AI integration tests by alexander-alderman-webb in #5347

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