docs(guide): document Cheaper Inference as an openai_compatible provider example

[aiapienthusiast]

Summary

This PR adds Cheaper Inference to the openai_compatible examples in docs/guide/new-backend.md.
It is docs only.
It uses the same pattern as #217.

Cheaper Inference is an OpenAI-compatible LLM gateway.

Cheaper Inference is one of the fastest-growing AI routers.
Each model costs 15–60% less than the list price of its lab.

Details

Changes:

• One row in the provider table.
• One short paragraph under the table.

Tests:

• python -m mkdocs build --strict: passed.

Usage:

export OPENAI_COMPATIBLE_BASE_URL="https://api.cheaperinference.com/v1"
export OPENAI_COMPATIBLE_API_KEY="ci_live_..."
export OPENAI_COMPATIBLE_MODEL="gpt-5.4-mini"

[Yifan Yang (Yif-Yang)]

Thank you for contributing a provider configuration example. Before adding another vendor-specific entry, could you explain the distinct user need it addresses that the existing generic openai_compatible configuration and gateway examples do not already cover? A concrete provider-specific configuration requirement or documented interoperability detail would make its project value easier to assess.

Please also keep the guide neutral: remove the blanket “Each model costs 15–60% less” statement and the unsupported growth claim in the PR description. Pricing and catalog availability change independently of SkillOpt. If the example is retained, link API/model documentation supporting the base URL and model ID. No paid integration run or positive performance benchmark is requested.