Make Nx the primary build workflow for the browser extension docs

[sognefej]

🎟️ Tracking

📔 Objective

Updates the browser extension build instructions to make npx nx serve browser the recommended
build path instead of the legacy npm run build:watch scripts, matching the Nx migration already
called out in clients. The old npm-script instructions are preserved in a new "Manual build
commands" section rather than removed outright.

📸 Screenshots

N/A — documentation-only change.

[github-actions]

🤖 Bitwarden Claude Code Review

Overall Assessment: APPROVE

Reviewed a documentation-only change to docs/getting-started/clients/browser/index.md that promotes npx nx serve browser as the recommended build workflow and preserves the legacy npm-script instructions in a new "Manual build commands" section. Verified the Nx commands, output paths, and the #testing-and-debugging anchor reference are internally consistent across the Build instructions and Testing/debugging sections. The change follows the repository style guide and matches the Nx migration described in the PR objective. No security, correctness, or breaking-change concerns were found.

Code Review Details

No blocking findings.

Minor observation (not blocking): the "Manual build commands" section notes the npm scripts output to the build folder, while the Testing and debugging sections instruct opening dist/apps/browser/<configuration>. A reader who uses the manual path would need to substitute the folder accordingly. Acceptable given the manual section is explicitly de-emphasized.