[Bug] Chrome extension display name not synced in list_extensions response

Bug Description
Follow-up to my report on "Browser N" names in /chrome. ~/.claude.json has chromeExtension.pairedDeviceName = "2019PC" for deviceId e4548f7d-ba10-4df3-bd44-a509443c7a34. So the pairing_response round trip carries the name, but list_extensions results show it as "Browser 2". Claude Code 2.1.289 parses list_extensions entries with name: v(e.name, 50) only, and falls back to "Browser ${i+1}" when that is empty. The extension (1.0.98) sends the name as display_name in its connect message. Possible field-name mismatch between the bridge's list_extensions payload and the Claude Code parser, or the bridge not storing display_name from connect. Separately, the extension never sets bridgeDefaultDisplayName from the native host's computer_name on this machine.

Environment Info

• Platform: win32
• Terminal: windows-terminal
• Version: 2.1.289
• Feedback ID: 2bfa73a8-b9a0-4683-ac0f-a2dec5dd9601