refactor: introduce public/private headers

[dmehala]

Description

This refactor organizes the library headers into public and private categories. Public headers are now exposed in the include directory for external use, while private headers are kept within src directory.

Motivation

To establish a clear separation between the public API and internal code, enhancing maintainability and simplifying the usage of the library.

Additional Notes

The current refactor is still a work in progress. As you can see there's already a lot of changes and I did not want to push a huge PR too difficult to review. I will gradually improve that separation with subsequent PR.

Here's an exhaustive list of remaining work:

• Remove rate.h from the public API.
• Remove json_fwd.h from the public API.
• Remove sampling_decision.h from the public API.
• Remove sampling_mechanism.h from the public API.
• Remove span_defaults.h from the public API.
• Remove span_matcher.h from the public API.
• Remove trace_segment.h from the public API. (trace_segment is the trace context. Probably worth a renaming).
• Fix the JSON configuration.
• Run include-what-you-use.

[dmehala]

Is it still needed?

[codecov-commenter]

Codecov Report

Attention: Patch coverage is 93.75000% with 2 lines in your changes missing coverage. Please review.

Project coverage is 94.46%. Comparing base (aa037a9) to head (a8ba0a3).

Files	Patch %	Lines
src/datadog/remote_config/product.cpp	77.77%	2 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #144   +/-   ##
=======================================
  Coverage   94.46%   94.46%           
=======================================
  Files          73       72    -1     
  Lines        3811     3797   -14     
=======================================
- Hits         3600     3587   -13     
+ Misses        211      210    -1     

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