Update Lambda Trace Context Propagation Priority to Use New Lambda Context API

Is your feature request related to a problem? Please describe.

In traditional AWS Lambda functions, trace context propagation relies on extracting context from environment variables or system properties.
With the introduction of AWS Lambda Managed Instance, a single runtime instance may process multiple requests rather than just one. To support correct context propagation in this model, AWS Lambda has added a new Lambda API (available in updated Lambda runtime versions) that allows functions to retrieve the upstream trace context directly from the Lambda context.

To ensure accurate and reliable trace propagation across requests, the Lambda instrumentation needs to update its propagation logic.

Describe the solution you'd like

Update the context propagation priority in the AWS Lambda instrumentation as follows:

1. Use the new Lambda context API, if available, to extract the upstream trace context
2. Fallback to system properties, if present
3. Finally fallback to environment variables, as the last resort

This ensures correct context propagation for both traditional single-invoke runtimes and the new Managed Instance model.

Technical details

The new API is introduced in
https://github.com/aws/aws-lambda-java-libs/pull/563

Bump dependency for the new API:

library("com.amazonaws:aws-lambda-java-core:1.4.0")

API:

String com.amazonaws.services.lambda.runtime.Context.getTraceHeader()

Refer to OpenTelemetry .Net https://github.com/open-telemetry/opentelemetry-dotnet-contrib/pull/3410, AWS Lambda .Net has only .Net 8 one runtime version, it does not have to fallback to the environment variable.

Describe alternatives you've considered

No response

Additional context

No response

Tip

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Start in the AWS Lambda instrumentation and inspect the existing trace-context propagation logic and its current dependency declaration. Update the dependency to com.amazonaws:aws-lambda-java-core:1.4.0, use Context.getTraceHeader() before system properties and environment variables, and verify the priority with the relevant Lambda instrumentation tests.