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)

[yury-s]

Let's resolve the conflicts and we can merge this.

[willcoliveira]

Let's resolve the conflicts and we can merge this.

@yury-s , done. Thanks for the heads up.

[github-actions]

Test results for "tests 1"

4 failed
❌ [playwright-test] › playwright.trace.spec.ts:137 › should not throw with trace: on-first-retry and two retries in the same worker @macos-latest-node20
❌ [playwright-test] › reporter-line.spec.ts:193 › merged › should show error context with relative path @macos-latest-node20
❌ [playwright-test] › reporter.spec.ts:251 › created › should not have internal error when steps are finished after timeout @macos-latest-node20
❌ [playwright-test] › reporter.spec.ts:251 › merged › should not have internal error when steps are finished after timeout @macos-latest-node20

2 flaky

⚠️ [chromium-library] › library/trace-viewer.spec.ts:1216 › should display language-specific locators `@chromium-ubuntu-22.04-node20`
⚠️ [playwright-test] › ui-mode-trace.spec.ts:700 › should indicate current test status `@macos-latest-node20`

38657 passed, 841 skipped

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Merge workflow run.

[github-actions]

Test results for "MCP"

2 failed
❌ [chrome] › mcp/http.spec.ts:199 › http transport browser lifecycle (persistent) @mcp-macos-15
❌ [chrome] › mcp/launch.spec.ts:116 › isolated context @mcp-macos-15

5131 passed, 171 skipped

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Merge workflow run.

[dgozman]

Thank you for the PR!