feat: add C language binding

[adeshkumar1]

Description

Add C language binding (ddog_trace_*) wrapping the C++ tracing library for integration from C-based projects.
Follows Datadog's ddog_ naming convention from libdatadog. Gated behind BUILD_C_BINDING=OFF.

Motivation

Enable C-based projects to use dd-trace-cpp without a C++ dependency at the call site.

Building and Testing

cmake -B build -DBUILD_C_BINDING=ON -DDD_TRACE_BUILD_TESTING=ON
cmake --build build -j
./build/binding/c/test_c_binding

[pr-commenter]

Benchmarks

Benchmark execution time: 2026-03-04 22:27:52

Comparing candidate commit 527d3e2 in PR branch adeshkumar1/c-binding-on-main with baseline commit f8c3913 in branch main.

Found 0 performance improvements and 0 performance regressions! Performance is the same for 1 metrics, 0 unstable metrics.

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 87.76%. Comparing base (f8c3913) to head (527d3e2).

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #286   +/-   ##
=======================================
  Coverage   87.76%   87.76%           
=======================================
  Files          84       84           
  Lines        5658     5658           
=======================================
  Hits         4966     4966           
  Misses        692      692           

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[datadog-official]

🎯 Code Coverage (details)
• Patch Coverage: 100.00%
• Overall Coverage: 87.77% (+0.00%)

_{This comment will be updated automatically if new data arrives.
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[xlamorlette-datadog]

The instructions, given in the PR description, to build and test work but they create files everywhere.
Rather use the build directory, as in https://github.com/DataDog/dd-trace-cpp/blob/main/README.md#building.
So, that would be:

cmake -B build -DBUILD_C_BINDING=ON -DDD_TRACE_BUILD_TESTING=ON .
cmake --build build -j
./build/binding/c/test_c_binding

Please add these instructions in a new readme file in the C binding directory.