ToolHive always shows Amp clients as installed

[keeganwitt]

Bug description

ToolHive shows Amp as being installed, when it isn't.

Steps to reproduce

Run thv client status when you have VS Code, Cursor, or Windsurf IDEs installed.

Expected behavior

Amp clients are not shown as installed.

Actual behavior

ToolHive shows Amp clients as installed.

┌────────────────────┬───────────┬────────────┐
│ CLIENT TYPE        │ INSTALLED │ REGISTERED │
├────────────────────┼───────────┼────────────┤
│ roo-code           │ ✅ Yes    │ ✅ Yes    │
│ cline              │ ✅ Yes    │ ✅ Yes    │
│ vscode-insider     │ ❌ No     │ ❌ No     │
│ vscode             │ ✅ Yes    │ ✅ Yes    │
│ cursor             │ ✅ Yes    │ ✅ Yes    │
│ claude-code        │ ❌ No     │ ❌ No     │
│ windsurf           │ ✅ Yes    │ ✅ Yes    │
│ windsurf-jetbrains │ ✅ Yes    │ ✅ Yes    │
│ amp-cli            │ ❌ No     │ ❌ No     │
│ amp-vscode         │ ✅ Yes    │ ✅ Yes    │
│ amp-vscode-insider │ ❌ No     │ ❌ No     │
│ amp-cursor         │ ✅ Yes    │ ✅ Yes    │
│ amp-windsurf       │ ✅ Yes    │ ✅ Yes    │
│ lm-studio          │ ✅ Yes    │ ✅ Yes    │
│ goose              │ ❌ No     │ ❌ No     │
└────────────────────┴───────────┴────────────┘

Environment (if relevant)

• ToolHive version: 0.3.7

Additional context

• https://github.com/stacklok/toolhive/blob/c2af88da4a20ea86c6a11e05442d6843ff311002/pkg/client/config.go looks for amp.mcpServers in settings.json of these VS Codium clients.
• Manually deleting the Amp clients from %AppDataLocal%\ToolHive\groups\default.json didn't help.