This commit updates all references throughout the codebase from "Claude Code
Proxy API" to the shorter "CCProxy API" branding. The changes include:

- Updated project name in documentation files (README, CONTRIBUTING, docs/)
- Updated module docstrings and comments across all Python files
- Updated CLI help text and configuration file headers
- Updated Docker service names in examples and documentation
- Updated GitHub repository URLs to use ccproxy-api instead of
  claude-code-proxy-api
- Added clarification in README that ccproxy is for Anthropic Claude LLM
- Updated all systemd service references and Docker container names

No functional changes were made - this is purely a branding/naming update
to provide a more concise project identity while maintaining clarity about
the project's purpose as a proxy for Claude API access.