docs: update README and lib.rs documentation

[alekseysidorov]

First part of #34 plan.

[alekseysidorov]

Merged ✅ — here's a summary of what this PR covers and what's left from #34 acceptance criteria.

✅ Already done (in this PR)

• New README structure: description → "Why context-logger?" → Usage → Async → License
• Clear positioning vs tracing ("Is this a replacement for tracing?" section)
• Cargo.toml description updated to match 0.2 model
• Updated crate-level docs in src/lib.rs with "How it works" + "Compatibility" sections
• All unit tests (17), integration tests (4 suites), and doc-tests (15) passing

📋 Still remaining for #34

Medium priority:

1. Add a standalone "Example output" block near the top of README.
   Currently the example output is inline in Rust comments (// service=api timestamp=...). Issue Rewrite README and docs for 0.2 release #34 asks for a visible terminal-style block:

   ### Example output
   
   ```text
   INFO Processing request service=api version=1.2.3 request_id=req-123 user_id=42
   INFO Fetching preferences service=api version=1.2.3 request_id=req-123 operation=fetch_preferences

2. Show both with_inherited_record and with_local_record in examples.
   The README currently only demonstrates with_inherited_record. Need an example that shows with_local_record for contrast (request-level vs operation-level fields).

Low priority:

3. Clean up indentation in doc-examples. Some Rust code blocks still have inconsistent 5–6 space indents instead of standard 4-space alignment.
4. Add a brief description to ContextLogger struct docs. Currently just "See the crate-level docs" — add at least one sentence about what it wraps.

Not covered by this PR (separate issue/PR):

5. Update examples/ — ensure any crate examples in examples/ directory reflect the 0.2 API and compile.
6. Prepare changelog for 0.2 release. Keep it short and scannable per Rewrite README and docs for 0.2 release #34 requirements.