feat(llmo): Make it optional to pass posthog client

[Twixes]

It's been inconvenient to have to explicitly pass a posthog_client to every AI provider wrapper. We should do what happens when you call posthog methods like posthog.capture() – run setup() to init the default_client and use default_client.

This is what this PR introduces.

[greptile-apps]

Greptile Summary

This PR modifies all AI provider wrapper classes (OpenAI, Anthropic, and their variants) to make the posthog_client parameter optional in their constructors. Instead of requiring explicit client initialization, the wrappers now follow PostHog's standard pattern where if no client is provided, they automatically call posthog.setup() to initialize and use the default_client. The change affects all AI provider implementations including async versions.

This improvement aligns the AI wrappers with PostHog's core functionality pattern (like posthog.capture()), making the library more ergonomic while maintaining the flexibility to use custom clients when needed.

Confidence score: 5/5

1. This PR is very safe to merge as it adds convenience without breaking existing functionality
2. The implementation is consistent across all providers, uses proper type safety with cast(), and follows established PostHog patterns
3. Key files that warrant attention:
   • posthog/ai/openai/openai.py
   • posthog/ai/anthropic/anthropic.py
   • Corresponding async implementations

_{6 files reviewed, 3 comments}

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[k11kirky]

Looks good. Added to gemini as well, Realized i need to fix the docs as well and support this for node.