feat(mcp): pass MCP client name to extension connection

[yury-s]

Summary

• Add clientName to ClientInfo; MCP server populates it from the connecting client's name (via getClientVersion()), defaulting to 'Playwright MCP'
• CLI daemon guesses the name from env vars (CLAUDECODE → 'Claude Code', COPILOT_CLI → 'GitHub Copilot'), defaulting to 'playwright-cli'
• Pass the client name to the extension browser connection, replacing the hardcoded 'Playwright Agent'
• Remove unused exports (createClientInfo, startCliDaemonServer, logUnhandledError, toMcpTool) from exports.ts

[pavelfeldman]

please fix what agreed offline

[github-actions]

Test results for "tests 1"

1 failed
❌ [playwright-test] › ui-mode-test-network-tab.spec.ts:397 › should not preserve selection across test runs @macos-latest-node20

6 flaky

⚠️ [chromium-library] › library/popup.spec.ts:261 › should not throw when click closes popup `@chromium-ubuntu-22.04-arm-node20`
⚠️ [chromium-library] › library/video.spec.ts:342 › screencast › should work for popups `@chromium-ubuntu-22.04-node24`
⚠️ [chromium-library] › library/video.spec.ts:724 › screencast › should work with video+trace `@chromium-ubuntu-22.04-node20`
⚠️ [chromium-library] › library/popup.spec.ts:261 › should not throw when click closes popup `@chromium-ubuntu-22.04-node22`
⚠️ [firefox-library] › library/inspector/cli-codegen-1.spec.ts:1080 › cli codegen › should not throw csp directive violation errors `@firefox-ubuntu-22.04-node20`
⚠️ [playwright-test] › ui-mode-test-update.spec.ts:162 › should pick new / deleted nested tests `@windows-latest-node20`

39145 passed, 846 skipped

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Merge workflow run.

[github-actions]

Test results for "MCP"

2 failed
❌ [chrome] › mcp/dashboard.spec.ts:23 › should show browser session chip @mcp-ubuntu-latest
❌ [msedge] › mcp/files.spec.ts:106 › clicking on download link emits download @mcp-windows-latest

6427 passed, 383 skipped

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Merge workflow run.