[sdk-logs] Update ILogger integration to use LoggerProvider

[CodeBlanch]

Relates to #4433

Changes

• Updates the ILogger integration inside SDK to use the LoggerProvider bits

Merge requirement checklist

• CONTRIBUTING guidelines followed (nullable enabled, static analysis, etc.)
• Unit tests added/updated

[codecov]

Codecov Report

Merging #4473 (d6642cc) into main (fea6793) will increase coverage by 0.07%.
The diff coverage is 91.11%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4473      +/-   ##
==========================================
+ Coverage   85.02%   85.10%   +0.07%     
==========================================
  Files         314      314              
  Lines       12524    12524              
==========================================
+ Hits        10649    10659      +10     
+ Misses       1875     1865      -10     

Impacted Files	Coverage Δ
src/OpenTelemetry/ProviderExtensions.cs	91.66% <60.00%> (-3.79%)	⬇️
.../OpenTelemetry/Logs/ILogger/OpenTelemetryLogger.cs	87.73% <87.50%> (-0.02%)	⬇️
...emetry/Logs/ILogger/OpenTelemetryLoggerProvider.cs	95.16% <97.29%> (+4.07%)	⬆️
...lemetry/Logs/ILogger/OpenTelemetryLoggerOptions.cs	85.00% <100.00%> (-15.00%)	⬇️
...try/Logs/ILogger/OpenTelemetryLoggingExtensions.cs	100.00% <100.00%> (ø)

... and 7 files with indirect coverage changes