fix(caching): honor supported_call_types for embeddings and router functions (#41003) - #41008

TLDR

Problem this solves:

• Embedding calls were still being cached when supported_call_types excluded embedding/aembedding.
• Router call types starting with an underscore (e.g. _aembedding) failed exact string match against supported_call_types.

How it solves it:

• Enforce supported_call_types in Cache.should_use_cache and caching handler methods.
• Match candidate function/call types with and without leading underscores (.lstrip("_")).
• Pass call_type and original_function into cache lookup and pipeline write tasks.

User Flow

Before: setting supported_call_types: ["completion", "acompletion"] continues to cache /v1/embeddings requests.

1. Proxy admin configures cache_params.supported_call_types: ["completion", "acompletion"] to only cache chat completions.
2. User sends POST /v1/embeddings with input "hello".
3. Gateway stores embeddings in cache and subsequent calls return cached embedding responses instead of executing upstream.

After: embedding requests bypass cache completely when excluded from supported_call_types.

1. Proxy admin configures cache_params.supported_call_types: ["completion", "acompletion"].
2. User sends POST /v1/embeddings with input "hello".
3. Gateway does not look up or write embeddings into cache; calls are executed upstream as intended.

Relevant issues

Fixes #41003

Pre-Submission checklist

• I have added meaningful tests
• The handful of test files covering my change pass locally, e.g. uv run pytest tests/test_litellm/<your_test_file>.py -v.
• My PR passes all required CI/CD checks
• My PR's scope is as isolated as possible; it only solves 1 specific problem
• I have received a Greptile Confidence Score of at least 4/5 before requesting a maintainer review

Type

🐛 Bug Fix

Final Attestation

• The tests check the right things, including the edge cases, and regressions in the respective real-world customer use-cases are not possible after this PR