[FEATURE] support TRACEPARENT context propagation for OpenTelemetry

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

We use opentelemetry to expose metrics, logs and traces of our developer environments and github action workflows. All our developer environments have a TRACEPARENT= environment variable set (https://opentelemetry.io/docs/specs/otel/context/env-carriers/)
that tracks the trace for a certain shell session.

We want to enable Claude Code's telemetry integration (https://code.claude.com/docs/en/monitoring-usage)
but we want it to integrate with our existing telemetry stack; so that we can correlate claude code sessions to the developer environments that they're spawned in

Proposed Solution

claude should read the TRACEPARENT= env var and use that to set the traceId and spanId field on the log records, and the span_id and trace_id examplars on the metric events, so that telemetry that claude generates can be correlated to the trace/span in which they were generated.

Alternative Solutions

None

Priority

High - Significant impact on productivity

Feature Category

Developer tools/SDK

Use Case Example

_No response_

Additional Context

_No response_