Motivation

• Ensure embedded agent lifecycle debug logs always include the model and provider context so operator logs show what model/provider were used for a run, and log the runtime label when the runtime is claude-sdk.

Description

• Add optional provider, model, and runtime fields to SubscribeEmbeddedPiSessionParams so subscribing handlers can access run context (src/agents/pi-embedded-subscribe.types.ts).
• Update lifecycle handlers to append a formatted context string (provider, model, and runtime when runtime === "claude-sdk") to embedded run agent start and embedded run agent end debug logs (src/agents/pi-embedded-subscribe.handlers.lifecycle.ts).
• Wire the embedded runner to pass provider, modelId, and a runtime label into subscribeEmbeddedPiSession so the lifecycle logger receives the values at run time (src/agents/pi-embedded-runner/run/attempt.ts).

Testing

• Ran the repository format check pnpm -s check, which initially reported formatting issues; then fixed formatting with pnpm -s oxfmt --write and verified formatting with pnpm -s oxfmt --check, which succeeded.
• Attempted to run the related e2e test with pnpm -s vitest run src/agents/pi-embedded-subscribe.lifecycle-billing-error.e2e.test.ts, but the repository Vitest config excludes *.e2e.test.ts files so that invocation did not execute (no test files found in the configured include patterns).

Codex Task