* fix: Fixes https://github.com/BerriAI/litellm/issues/23185
* fix(responses/main.py): ensure litellm metadata custom cost works
* refactor: move all logging updates to a common function, to have just 1 place to update logging kwarg updates
image_edit was not forwarding model_info/metadata to the logging object,
so custom_pricing was never detected. After PR #20679 stripped custom
pricing fields from the shared backend key, image_edit cost became 0.
Fixes#22244
Fixes#22285 — extra_headers passed to litellm.image_generation() were
silently dropped on the openai/litellm_proxy/openai_compatible_providers
code path. The azure and azure_ai paths already forwarded them correctly.
