Add API bindings section to root README

[trmartin4]

📔 Objective

Adds documentation to the root README to describe how to consume and update API bindings.

Includes changes from #622 in which @harr1424 identified issues with our local binding generation instructions as well. Despite this not being the preferred way to update bindings, it's important for those instructions to be accurate as well.

Corresponding PR for removing the docs from Contributing Docs: bitwarden/contributing-docs#735.

Related PR for adding more documentation to bitwarden-core: #618.

🚨 Breaking Changes

⏰ Reminders before review

• Contributor guidelines followed
• All formatters and local linters executed and passed
• Written new unit and / or integration tests where applicable
• Protected functional changes with optionality (feature flags)
• Used internationalization (i18n) for all UI strings
• CI builds passed
• Communicated to DevOps any deployment requirements
• Updated any necessary documentation (Confluence, contributing docs) or informed the documentation
  team

🦮 Reviewer guidelines

• 👍 (:+1:) or similar for great changes
• 📝 (:memo:) or ℹ️ (:information_source:) for notes or general info
• ❓ (:question:) for questions
• 🤔 (:thinking:) or 💭 (:thought_balloon:) for more open inquiry that's not quite a confirmed
  issue and could potentially benefit from discussion
• 🎨 (:art:) for suggestions / improvements
• ❌ (:x:) or ⚠️ (:warning:) for more significant problems or concerns needing attention
• 🌱 (:seedling:) or ♻️ (:recycle:) for future improvements or indications of technical debt
• ⛏ (:pick:) for minor or nitpick changes

[github-actions]

Checkmarx One – Scan Summary & Details – 37c6cb84-ce2a-4876-a2bf-9d1c7bd603a3

Great job! No new security vulnerabilities introduced in this pull request

[github-actions]

🔍 SDK Breaking Change Detection Results

SDK Version: doc-improvement-api-workflow-addition (de81279)
Completed: 2025-12-30 03:07:10 UTC
Total Time: 215s

Client	Status	Details
typescript	✅ No breaking changes detected	TypeScript compilation passed with new SDK version - View Details

---

Breaking change detection completed. View SDK workflow

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 78.71%. Comparing base (0839e61) to head (de81279).
⚠️ Report is 5 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #638   +/-   ##
=======================================
  Coverage   78.71%   78.71%           
=======================================
  Files         290      290           
  Lines       31266    31266           
=======================================
  Hits        24611    24611           
  Misses       6655     6655           

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[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud