### What does this PR do?

Implements `DD_TRACE_PROPAGATION_BEHAVIOR_EXTRACT` for dd-trace-go.

See [RFC](https://docs.google.com/document/d/1KWx5F-XEzJ55XZPwY30t0otVIyOUYHWsDhEt3YPDnso/edit?pli=1&tab=t.0).

Three modes:
- **`continue`** (default): existing behavior -- incoming trace context is used as the local trace context. Baggage is propagated.
- **`restart`**: starts a new trace with a fresh trace ID and sampling decision. The incoming context is referenced via a span link with `reason=propagation_behavior_extract`. Baggage is propagated.
- **`ignore`**: discards the entire incoming trace context. Returns `nil, nil` from `Extract()` so callers produce a fresh root span with no parent, no span links, and no baggage.

When `DD_TRACE_PROPAGATION_EXTRACT_FIRST=true` is combined with `restart`, baggage is explicitly extracted after the first propagator returns, since the RFC requires baggage to be propagated in `restart` mode regardless.

### Motivation

See RFC: [Trace Context Propagation Extraction Modes](https://docs.google.com/document/d/1htJFBWR4RFpmK1i6wopxkZwlvYu9unkPJnMnTom4Xjo)
System-tests PR: DataDog/system-tests#6757

### Reviewer's Checklist

- [x] Changed code has unit tests for its functionality at or near 100% coverage.
- [x] [System-Tests](https://github.com/DataDog/system-tests/) covering this feature have been added and enabled with the va.b.c-dev version tag.

Note: Still working on it: DataDog/system-tests#6757

- [ ] There is a benchmark for any new code, or changes to existing code.

Note: Benchmarking is explicitly out of scope per the RFC ("the processing overhead is minimal in comparison to the rest of the context extraction operation").

- [x] If this interacts with the agent in a new way, a system test has been added.
- [x] New code is free of linting errors. You can check this by running `make lint` locally.
- [x] New code doesn't break existing tests. You can check this by running `make test` locally.
- [x] Add an appropriate team label so this PR gets put in the right place for the release notes.
- [x] All generated files are up to date. You can check this by running `make generate` locally.
- [x] Non-trivial go.mod changes, e.g. adding new modules, are reviewed by @DataDog/dd-trace-go-guild. Make sure all nested modules are up to date by running `make fix-modules` locally.

Unsure? Have a question? Request a review!


Co-authored-by: genesor <[email protected]>
Co-authored-by: kakkoyun <[email protected]>

…l make_distant_call

PR #6757 added span-context injection and header-key lowercasing to the
net-http /make_distant_call handler and activated
Test_ExtractBehavior_Restart_With_Extract_First at v2.10.0-dev for all Go
weblogs. The net-http-span-pool clone never received the matching handler
change, so after merging main the test ran against span-pool and failed with
KeyError: 'x-datadog-trace-id'.

Mirror the net-http handler: inject the active span context into the outbound
request headers and lowercase the captured request-header keys.

Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>