[DOCS] Document the AI_AGENT environment variable

Documentation Type

Missing documentation (feature not documented)

Documentation Location

https://code.claude.com/docs/en/env-vars

Section/Topic

AI_AGENT environment variable

What's Wrong or Missing?

Since 2.1.120, Claude Code sets AI_AGENT in subprocesses (e.g. claude-code_2-1-284_agent), but it's only mentioned in the changelog.

It's now the cross-vendor agent marker: VS Code Copilot, pi, OpenCode, and others set it, and tools like gh, Jest, Vitest, and Composer change behavior based on it. Some consumers match the exact value, and the version suffix already breaks them (herdrdev/herdr#4631).

Suggested Improvement

Add an AI_AGENT row next to CLAUDECODE that covers:

1. The value format, and which parts are stable
2. Which subprocesses receive it (tools, hooks, MCP servers, IDE terminals)
3. Whether an AI_AGENT value inherited from an outer agent is kept or overwritten

Previously requested in #53071 (stale-closed). Cross-vendor context: agentsmd/agents.md#136.

Impact

Medium - Makes feature difficult to understand