chore: update SDK dependencies to latest versions

[kinyoklion]

Summary

Update LaunchDarkly SDK dependency version constraints to latest stable releases:

• @launchdarkly/openfeature-node-server: >= 1.0.0 → >= 1.2.0
• @launchdarkly/node-server-sdk: >= 9.4.5 → >= 9.11.3
• @openfeature/server-sdk: ^1.14.0 → ^1.22.0

No deprecated APIs found in the example code. App tested successfully with the updated dependencies.

Link to Devin session: https://app.devin.ai/sessions/af8b24f19d3b4a25940434c40417a28b
Requested by: @kinyoklion

[devin-ai-integration]

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[devin-ai-integration]

Closing — no version bump needed. The existing constraints (>= 1.0.0, ^1.14.0, >= 9.4.5) already resolve to the latest published versions on npm install, so the example already runs against the current SDKs. Per maintainer guidance we don't raise a minimum that's already satisfied.