fix(mcp-sync): route tests never follow COPILOT_HOME, and docs name Copilot (#581)

The route test cleared only the registry CLIs' relocation vars, so with
COPILOT_HOME exported it wrote its fixture into that real Copilot config.
It now clears the sync-only tools' vars too, and Copilot's install probe goes
through the test's own installed set instead of the machine's PATH.

Docs: CLAUDE.md, the API reference and the Settings reference name COPILOT_HOME
and the sync-only table; a missing comma in docs/cli-registry.md; the
mcp-sync.ts overview and the Sync confirm mention Copilot.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

docs/api-reference.md

@@ -836,7 +836,7 @@ Result (`data`):
 - `targets[]` — one per enabled CLI that declares an MCP config, plus GitHub Copilot CLI (`id: "copilot"`, a sync-only target that is not a run mode): `id`, `label`, `file`, `status`, `error?`, `servers` (names it already has), `added` (names added, or that would be), `skipped` (names its dialect cannot express, e.g. SSE for Codex and Antigravity).
   - `status`: `ok`; `absent` (not installed and no config file, so not read or created); `skipped` (the CLI's relocation env var, e.g. `CODEX_HOME`, is set to a relative path in the server's environment, so its file cannot be located safely and is neither read nor written); `unreadable` (the file exists but cannot be parsed safely, so it is not written); `failed` (a read or write error, the file may be unchanged).
   - `error` says why a target is not `ok`. A parse failure is reported by position only (`not valid TOML (line 3, column 21)`, `not valid JSON`), never with text from the file.
-  - `file` honours each CLI's own relocation env var as the server process sees it (`CLAUDE_CONFIG_DIR`, `CODEX_HOME`, `XDG_CONFIG_HOME`, `GEMINI_CLI_HOME`); see `docs/cli-registry.md`.
+  - `file` honours each CLI's own relocation env var as the server process sees it (`CLAUDE_CONFIG_DIR`, `CODEX_HOME`, `XDG_CONFIG_HOME`, `GEMINI_CLI_HOME`, and `COPILOT_HOME` for the sync-only Copilot CLI); see `docs/cli-registry.md`.
 - `conflicts[]` — names defined differently by different CLIs. Existing definitions are kept; the first CLI's is copied where the name is missing.
 - `disabled[]` — names left out because every definition is switched off in its own CLI (codex `enabled = false`, opencode `enabled: false`, antigravity `disabled: true`).
 - `unsupported[]` — labels of enabled agent CLIs with no known MCP config file (nothing is guessed).