docs: fix deprecated Playwright API usage in running-code.md

[willcoliveira]

Summary

• Replace deprecated page.click(), page.fill(), page.waitForSelector() with modern role locators equivalents in running-code.md
• Fix Promise.all race condition anti-pattern in file download example
• @yury-s fix: update deprecated Playwright API usage in running-code docs playwright-cli#277 ✅

Test plan

• Verify all code examples in running-code.md use current Playwright APIs
• Confirm no deprecated methods remain (page.click, page.fill, page.waitForSelector)

[github-actions]

Test results for "tests 1"

2 failed
❌ [playwright-test] › expect.spec.ts:562 › should support toHaveURL predicate @macos-latest-node20
❌ [playwright-test] › expect.spec.ts:964 › should support mergeExpects @macos-latest-node20

3 flaky

⚠️ [chromium-library] › library/inspector/recorder-api.spec.ts:120 › should type `@chromium-ubuntu-22.04-node22`
⚠️ [firefox-library] › library/beforeunload.spec.ts:188 › does not get stalled by beforeUnload `@firefox-ubuntu-22.04-node20`
⚠️ [playwright-test] › ui-mode-trace.spec.ts:185 › should show snapshots for steps `@macos-latest-node20`

18607 passed, 433 skipped

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Merge workflow run.

[github-actions]

Test results for "MCP"

4 failed
❌ [chrome] › mcp/mouse.spec.ts:75 › browser_mouse_click_xy (right button) @mcp-windows-latest
❌ [chromium] › mcp/mouse.spec.ts:75 › browser_mouse_click_xy (right button) @mcp-windows-latest
❌ [msedge] › mcp/mouse.spec.ts:75 › browser_mouse_click_xy (right button) @mcp-windows-latest
❌ [webkit] › mcp/http.spec.ts:199 › http transport browser lifecycle (persistent) @mcp-macos-15

5038 passed, 171 skipped

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Merge workflow run.