LCORE-3312: Refactored exception handling

[asimurka]

Description

Migrates OGX error handling from the legacy APIConnectionError / APIStatusError / OpenAI-style status exceptions to the unified ogx_client.ApiException model (status=None for connection failures, HTTP status otherwise), and updates endpoints, utilities, and unit/integration/e2e tests to match. Also tightens a few related paths (prompts/providers/conversations) and keeps Singleton typing compatible with mypy and pydocstyle.

Type of change

• Refactor
• New feature
• Bug fix
• CVE fix
• Optimization
• Documentation Update
• Configuration Update
• Bump-up service version
• Bump-up dependent library [pyproject.toml + uv.lock]
• Bump-up dependent library [requirements.*.txt for Konflux]
• Bump-up library or tool used for development (does not change the final image)
• CI configuration change
• Konflux configuration change
• Unit tests improvement
• Integration tests improvement
• End to end tests improvement
• Benchmarks improvement

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Related Tickets & Documents

• Related Issue #
• Closes # https://redhat.atlassian.net/browse/LCORE-3312

Checklist before requesting a review

• I have performed a self-review of my code.
• PR has passed all pre-merge test jobs.
• If it is a core feature, I have added thorough tests.

Testing

• Please provide detailed steps to perform tests related to this code change.
• How were the fix/results from this change verified? Please provide relevant screenshots or results.

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