docs: centralize SDK examples in official docs

[marandaneto]

💡 Motivation and Context

SDK README and usage markdown files duplicated code snippets that can drift from the official documentation. This keeps public SDK usage examples in the official docs as the single source of truth while preserving local repository information like badges, Questions/community links, and example setup instructions.

💚 How did you test it?

• Ran git diff --check.
• Verified stale USAGE.md/old integration-doc links were removed or updated.
• Verified existing badges and Questions/community sections are preserved.

📝 Checklist

• I reviewed the submitted code.
• I added tests to verify the changes.
• I updated the docs if needed.
• No breaking change or entry added to the changelog.

If releasing new changes

• Ran pnpm changeset to generate a changeset file

[greptile-apps]

Prompt To Fix All With AI

Fix the following 1 code review issue. Work through them one at a time, proposing concise fixes.

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### Issue 1 of 1
README.md:1
**CI badge removed beyond stated PR scope**

The PR description focuses on removing duplicated code snippets, but the CI status badge (`[![CI](https://github.com/PostHog/posthog-php/actions/workflows/php.yml/badge.svg)](…)`) and the PHP version badge were also dropped. These badges give contributors an at-a-glance signal about build health without requiring a click through to GitHub Actions; their removal is unrelated to centralising documentation examples and may have been accidental.

_{Reviews (1): Last reviewed commit: "docs: centralize SDK examples in officia..." | Re-trigger Greptile}

[marandaneto]

Addressed the README feedback in 40ee1ab: restored the badges and added the Laravel docs link.

[greptile-apps]

_{Reviews (2): Last reviewed commit: "docs: address README review feedback" | Re-trigger Greptile}