chore(README): update links, add examples

[avivkeller]

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Project	Deployment	Preview	Updated (UTC)
api-docs-tooling	Ready	Preview	Oct 12, 2025 2:44pm

[Copilot]

Pull Request Overview

This PR updates the README.md file to reflect a repository rename from "api-docs-tooling" to "doc-kit" and adds comprehensive usage examples for the tool.

• Updates external links and badges to use the new repository name
• Adds an additional CLI invocation example
• Adds detailed examples section showing legacy and redesigned documentation generation workflows

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 74.55%. Comparing base (c73cdef) to head (0ea2fcd).
⚠️ Report is 2 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #454   +/-   ##
=======================================
  Coverage   74.55%   74.55%           
=======================================
  Files         107      107           
  Lines       10453    10453           
  Branches      681      681           
=======================================
  Hits         7793     7793           
  Misses       2658     2658           
  Partials        2        2           

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