fix(ai): preserve Gemini tool-call IDs in captures

[gouveags]

Problem

Gemini returns IDs for function calls, but @posthog/ai omits them from $ai_output_choices in both non-streaming and streaming generations. The live-provider cassettes in #5037 record call_649034 and call_388245; replaying them against the previous formatter reproduced the omission. This is the output-ID half of #5085.

Changes

• Preserve Gemini's function-call ID in both output paths when present. Calls without an ID keep their existing shape.
• Assert the IDs against the two recorded Gemini tool-call cassettes.
• Add an @posthog/ai patch changeset.

The replay assertion failed for both tool-call cases before the fix and passes afterward (5/5). The package build, type check, 1,024 unit tests, 275 offline cassette tests, and package lint pass. No provider request or response behavior changes.

The automated draft #5086 also covers this issue. This PR isolates the output-ID change and its cassette proof; the follow-up-input fix is being reviewed separately.

Release info Sub-libraries affected

Libraries affected

• @posthog/ai

Checklist

• Tests for new code
• Accounted for the impact of any changes across different platforms
• Accounted for backwards compatibility of any changes (no breaking changes!)
• Took care not to unnecessarily increase the bundle size
• Added an @posthog/ai patch changeset

🤖 Agent context

Autonomy: Human-driven (agent-assisted)

I directed the scope and reviewed the implementation with Codex. We used Vitest, the existing Gemini replay harness, and GitHub CLI. The first signed commit contains the failing test; the second contains the fix. No agent session link is shared because this private work session also includes unrelated workspace context. GitHub would not allow assigning my external account to this PR.

[gouveags]

@greptileai review

[greptile-apps]

The PR appears safe to merge.

_{Reviews (1) · Last reviewed commit: "fix(ai): retain Gemini tool call IDs in ..."}

[marandaneto]

The code looks good; no qualifying findings. Verdict: correct.