ci: lint and type-check the examples

[rqbazan]

Summary

Pull requests that touch the examples or the library now lint and type-check every example, with the same mise run examples:check used locally. The development and contributing docs list the new check.

 On pull requests touching examples/ or package/
   E2E        Playwright suite per example
+  Examples   lint and type-check every example

Evidence

• Before: no workflow runs examples:check, so a lint or type error in an example is not caught on a pull request.
  After: locally, mise run examples:check passes for every example, also with CI=1. A type error added to backend-node makes it fail with TS2322. On this PR, the "Check examples" check passes for every example.

Merge Danger

Door: two-way

Blast Radius: CI-only. "Check examples" is not a required check, like the docsite build in #51.

🤖 Generated with Claude Code

https://claude.ai/code/session_012xRhgX9oTNKCAhLoiC8vSX

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